Issue 1:
I checked it with 1.0.0 and 1.1.0-DEV.334.

julia> struct Spam end

julia> Base.show(::IO, ::Type{Spam}) = error("nope")

julia> Spam
Error showing value of type DataType:
ERROR: nope
Stacktrace:
[1] error(::String) at ./error.jl:33
[2] show(::IOContext{REPL.Terminals.TTYTerminal}, ::Type{SYSTEM: show(lasterr) caused an error
ErrorException("nope")

Stacktrace:
[1] error(::String) at ./error.jl:33
[2] show(::IOContext{REPL.Terminals.TTYTerminal}, ::Type{ERROR: nope
Stacktrace:
[1] error(::String) at ./error.jl:33
[2] show(::IOContext{REPL.Terminals.TTYTerminal}, ::Type{fatal: error thrown and no exception handler available.
ErrorException("nope")
rec_backtrace at /home/takafumi/repos/watch/julia/src/stackwalk.c:94
record_backtrace at /home/takafumi/repos/watch/julia/src/task.c:249
jl_throw at /home/takafumi/repos/watch/julia/src/task.c:580
error at ./error.jl:33
show at ./REPL[2]:1
jl_fptr_trampoline at /home/takafumi/repos/watch/julia/src/gf.c:1843
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
show_datatype at ./show.jl:526
show at ./show.jl:436
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
print at ./strings/io.jl:31
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
print at ./strings/io.jl:42
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
show_tuple_as_call at ./show.jl:1490
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
show_spec_linfo at ./stacktraces.jl:261
#show#9 at ./stacktraces.jl:272
unknown function (ip: 0x7f99e80ea7c3)
#show at ./none:0 [inlined]
#show_trace_entry#630 at ./errorshow.jl:471
unknown function (ip: 0x7f99e80ea0ec)
#show_trace_entry at ./none:0
unknown function (ip: 0x7f99e80e9e28)
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
show_backtrace at ./errorshow.jl:574
#showerror#612 at ./errorshow.jl:79
unknown function (ip: 0x7f99e80e7357)
jl_fptr_trampoline at /home/takafumi/repos/watch/julia/src/gf.c:1843
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
showerror at ./errorshow.jl:74 [inlined]
display_error at ./client.jl:99
jl_fptr_trampoline at /home/takafumi/repos/watch/julia/src/gf.c:1843
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
display_error at ./client.jl:102
jl_fptr_trampoline at /home/takafumi/repos/watch/julia/src/gf.c:1843
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
jl_apply at /home/takafumi/repos/watch/julia/src/julia.h:1559 [inlined]
jl_f__apply at /home/takafumi/repos/watch/julia/src/builtins.c:556
jl_f__apply_latest at /home/takafumi/repos/watch/julia/src/builtins.c:594
#invokelatest#1 at ./essentials.jl:697 [inlined]
invokelatest at ./essentials.jl:696 [inlined]
_start at ./client.jl:423
jl_apply_generic at /home/takafumi/repos/watch/julia/src/gf.c:2198
jl_apply at /home/takafumi/repos/watch/julia/ui/../src/julia.h:1559 [inlined]
true_main at /home/takafumi/repos/watch/julia/ui/repl.c:112
main at /home/takafumi/repos/watch/julia/ui/repl.c:233
__libc_start_main at /usr/lib/libc.so.6 (unknown line)
_start at ./julia (unknown line)

Issue 2:
jamie@machine:~/reltron/backend$ julia
_
_ _ _(_)_ | Documentation: https://docs.julialang.org
(_) | (_) (_) |
_ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help.
| | | | | | |/ _` | |
| | |_| | | | (_| | | Version 1.0.0 (2018-08-08)
_/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |

julia> versioninfo()
Julia Version 1.0.0
Commit 5d4eaca0c9 (2018-08-08 20:58 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: Intel(R) Xeon(R) CPU E3-1505M v5 @ 2.80GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-6.0.0 (ORCJIT, skylake)

(v1.0) pkg> activate ./

(Reltron) pkg> st
Project Reltron v0.1.0
Status `Project.toml`
[a93c6f00] DataFrames v0.13.1
[682c06a0] JSON v0.19.0
[0aa819cd] SQLite v0.6.0
[8dfed614] Test
[cf7118a7] UUIDs

julia> using DataFrames

julia> methods(hash)
# 62 methods for generic function "hash":
[1] hash(::Tuple{}, h::UInt64) in Base at tuple.jl:315
[2] hash(w::WeakRef, h::UInt64) in Base at hashing.jl:19
[3] hash(x::Expr, h::UInt64) in Base at hashing.jl:72
[4] hash(x::QuoteNode, h::UInt64) in Base at hashing.jl:73
[5] hash(x::UInt64, h::UInt64) in Base at float.jl:561
[6] hash(x::Int64, h::UInt64) in Base at float.jl:562
[7] hash(x::Float64, h::UInt64) in Base at float.jl:563
[8] hash(x::Float32, h::UInt64) in Base at float.jl:566
[9] hash(x::Char, h::UInt64) in Base at char.jl:196
[10] hash(r::Regex, h::UInt64) in Base at regex.jl:435
[11] hash(x::Base.Prehashed) in Base at multidimensional.jl:1307
[12] hash(v::VersionNumber, h::UInt64) in Base at version.jl:195
[13] hash(x::Cmd, h::UInt64) in Base at process.jl:70
[14] hash(x::Base.AndCmds, h::UInt64) in Base at process.jl:92
[15] hash(s::Base.SecretBuffer, h::UInt64) in Base at secretbuffer.jl:109
[16] hash(x::Float16, h::UInt64) in Base at hashing2.jl:169
[17] hash(frame::Base.StackTraces.StackFrame, h::UInt64) in Base.StackTraces at stacktraces.jl:90
[18] hash(a::Base.SHA1, h::UInt64) in Base at loading.jl:100
[19] hash(pkg::Base.PkgId, h::UInt64) in Base at loading.jl:169
[20] hash(id::LibGit2.GitHash, h::UInt64) in LibGit2 at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/LibGit2/src/oid.jl:185
[21] hash(cred::LibGit2.GitCredential, h::UInt64) in LibGit2 at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/LibGit2/src/gitcredential.jl:35
[22] hash(x::Dates.Year, h::UInt64) in Dates at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Dates/src/periods.jl:455
[23] hash(x::Dates.Month, h::UInt64) in Dates at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Dates/src/periods.jl:456
[24] hash(s::Random.DSFMT.DSFMT_state, h::UInt64) in Random.DSFMT at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Random/src/DSFMT.jl:42
[25] hash(r::Random.MersenneTwister, h::UInt64) in Random at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Random/src/RNGs.jl:162
[26] hash(r::Distributed.RRID, h::UInt64) in Distributed at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Distributed/src/messages.jl:15
[27] hash(r::Pkg.Types.VersionBound, h::UInt64) in Pkg.Types at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Pkg/src/versions.jl:88
[28] hash(s::Pkg.Types.VersionSpec, h::UInt64) in Pkg.Types at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Pkg/src/versions.jl:232
[29] hash(f::Pkg.Types.Fixed, h::UInt64) in Pkg.Types at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Pkg/src/Types.jl:85
[30] hash(i::Pkg.Pkg2.Pkg2Types.VersionInterval, h::UInt64) in Pkg.Pkg2.Pkg2Types at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Pkg/src/Pkg2/types.jl:25
[31] hash(s::Pkg.Pkg2.Pkg2Types.VersionSet, h::UInt64) in Pkg.Pkg2.Pkg2Types at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.0/Pkg/src/Pkg2/types.jl:158
[32] hash(s::DataStructures.IntSet, h::UInt64) in DataStructures at /home/jamie/.julia/packages/DataStructures/CImpO/src/int_set.jl:245
[33] hash(t::Tuple{Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Any,Vararg{Any,N}} where N, h::UInt64) in Base at tuple.jl:318
[34] hash(t::Tuple, h::UInt64) in Base at tuple.jl:316
[35] hash(p::Pair, h::UInt64) in Base at pair.jl:52
[36] hash(A::AbstractArray, h::UInt64) in Base at abstractarray.jl:2072
[37] hash(a::AbstractDict, h::UInt64) in Base at abstractdict.jl:462
[38] hash(x::NamedTuple, h::UInt64) in Base at namedtuple.jl:168
[39] hash(x::Union{Bool, Int16, Int32, Int8, UInt16, UInt32, UInt8}, h::UInt64) in Base at float.jl:565
[40] hash(z::Complex, h::UInt64) in Base at complex.jl:243
[41] hash(s::AbstractSet, h::UInt64) in Base at set.jl:292
[42] hash(x::AbstractChar, h::UInt64) in Base at char.jl:202
[43] hash(g::Base.Unicode.GraphemeIterator, h::UInt64) in Base.Unicode at strings/unicode.jl:671
[44] hash(ci::CartesianIndex, h::UInt64) in Base.IteratorsMD at multidimensional.jl:134
[45] hash(x::Rational{#s549} where #s549<:Union{Int16, Int32, Int64, Int8, UInt16, UInt32, UInt64, UInt8}, h::UInt64) in Base at hashing2.jl:147
[46] hash(x::Irrational, h::UInt64) in Base at irrationals.jl:127
[47] hash(x::Real, h::UInt64) in Base at hashing2.jl:32
[48] hash(s::Union{SubString{String}, String}, h::UInt64) in Base at hashing2.jl:177
[49] hash(x::Union{CategoricalString{R}, CategoricalValue{T,R} where T} where R, h::UInt64) in CategoricalArrays at /home/jamie/.julia/packages/CategoricalArrays/04bks/src/value.jl:135
[50] Error showing value of type Base.MethodList:
ERROR: UndefVarError: T not defined
Stacktrace:
[1] show(::IOContext{Base.GenericIOBuffer{Array{UInt8,1}}}, ::Type{SYSTEM: show(lasterr) caused an error
UndefVarError(:T)

Stacktrace:
[1] show(::IOContext{Base.GenericIOBuffer{Array{UInt8,1}}}, ::Type{ERROR: UndefVarError: T not defined
Stacktrace:
[1] show(::IOContext{REPL.Terminals.TTYTerminal}, ::Type{fatal: error thrown and no exception handler available.
UndefVarError(var=:T)
rec_backtrace at /buildworker/worker/package_linux64/build/src/stackwalk.c:94
record_backtrace at /buildworker/worker/package_linux64/build/src/task.c:246
jl_throw at /buildworker/worker/package_linux64/build/src/task.c:577
jl_undefined_var_error at /buildworker/worker/package_linux64/build/src/rtutils.c:134
show at /home/jamie/.julia/packages/WeakRefStrings/51hoN/src/WeakRefStrings.jl:56
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
show_datatype at ./show.jl:526
show at ./show.jl:436
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
print at ./strings/io.jl:31
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
print at ./strings/io.jl:42
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
show_tuple_as_call at ./show.jl:1490
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
show_spec_linfo at ./stacktraces.jl:261
#show#9 at ./stacktraces.jl:272
unknown function (ip: 0x7f8904210583)
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
#show at ./none:0 [inlined]
#show_trace_entry#631 at ./errorshow.jl:469
unknown function (ip: 0x7f890420fe4c)
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
#show_trace_entry at ./none:0
unknown function (ip: 0x7f890420fb48)
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
show_backtrace at ./errorshow.jl:572
#showerror#613 at ./errorshow.jl:79
unknown function (ip: 0x7f890420d1c7)
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
showerror at ./errorshow.jl:74 [inlined]
display_error at ./client.jl:99
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
display_error at ./client.jl:102
jl_fptr_trampoline at /buildworker/worker/package_linux64/build/src/gf.c:1829
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
jl_apply at /buildworker/worker/package_linux64/build/src/julia.h:1536 [inlined]
jl_f__apply at /buildworker/worker/package_linux64/build/src/builtins.c:556
jl_f__apply_latest at /buildworker/worker/package_linux64/build/src/builtins.c:594
#invokelatest#1 at ./essentials.jl:686 [inlined]
invokelatest at ./essentials.jl:685 [inlined]
_start at ./client.jl:423
jl_apply_generic at /buildworker/worker/package_linux64/build/src/gf.c:2182
unknown function (ip: 0x401ae8)
unknown function (ip: 0x401513)
__libc_start_main at /build/glibc-itYbWN/glibc-2.26/csu/../csu/libc-start.c:308
unknown function (ip: 0x4015b4)