Part I.

I'll try to give an answer and a response
more generally to the material in the OP.

Background. I got a BS in pure math with
nearly a second major in physics and
worked in computing and applied math for
problems in US national security around
DC. Jobs were very easy to get; at one
time my annual salary was 6 times what a
new high end Camaro cost; much of the work
was challenging for both the computing and
the applied math; I was learning a lot of
both computing, e.g., algorithms in Knuth,
and applied math on the job and also
especially math in independent study on
evenings and weekends. Soon I got a call
from a college friend to join FedEx. I
used my background in computing, with a
little applied math, to write some
software, in six weeks, to schedule the
fleet. The results pleased the BoD,
enabled crucial funding, and saved the
company. Later the BoD wanted some
revenue projections. I formulated and
solved

y'(t) = k y(t) (b - y(t))

for time t, revenue y(t) at time t, rate
of growth y'(t) = d/dt y(t) at time t, and
full revenue potential b. The BoD was
pleased, and ..., to make a long story
short, I saved the company again.

Then I got a Ph.D. in applied math from
the engineering school of a famous, high
end research university.

Now I'm doing an Internet startup, a Web
site, basically a new and very different
search engine -- for the, IMHO, very large
part of search handled at best poorly by
the existing Web sites and well known
techniques.

For the startup, my applied math
background is crucial: The crucial,
enabling core of the startup is some
original applied math I derived based on
some advanced pure math prerequisites I
got both in grad school and in independent
study. I already knew enough computing
except had to learn how to bring up a Web
site that has been easy for the user
interface but due to the core math
somewhat tricky on the server side. I
learned Microsoft's Visual Basic .NET (for
the programming language -- I like it),
ADO.NET (for the Web pages), and ASP.NET
for the (relatively meager) use of SQL
Server. The main difficulty was working
through 5000+ Web pages of documentation.
The first code is the first production
code, 100,000 lines of typing, 24,000
programming language statements, and lots
of documentation. The code seems to run
as intended, but I need to add some data.

Some lessons:

(1) Math. IMHO, the key to powerful,
valuable, new applications of computing is
applied math. That is, if we accept that
the big opportunity is to exploit and
apply current computing, then we might
notice that whatever we code to put out
data users will like, as information,
entertainment, whatever, is necessarily
mathematically something, understood or
not, powerful and valuable or not. So, in
some sense, from 100,000 feet up, it
should help to proceed mathematically,
with possibly advanced prerequisites, some
new results focused on the application in
mind, and complete with theorems and
proofs. Just IMHO. But I don't know
anyone with a yacht over 100' long that
did that; I suspect that very few people
agree with me. Maybe what I am saying is
a hopeless wild goose chase or a great
green field opportunity -- you judge.

(2) Getting a Ph.D. In a nutshell, the
three most important parts of getting a
Ph.D., in no particular order, are
research, research, and research. Yes,
there can be courses, credits, grades,
teaching assistant positions, weekly
research seminars, qualifying exams, etc.,
but at a research university what can "cut
through", dominate, and trump all or
nearly all of that is research. The
research should be publishable in a good
peer-reviewed journal of original
research; if there is any question, then
send it in.

The criterion for a Ph.D. dissertation may
be something like "An original
contribution to knowledge worthy of
publication." -- so, in case of some
doubt, publish the thing.

The usual criteria for publication are
that the work be (i) new, (ii) correct,
and (iii) significant.

Now for a non-standard observation and
recommendation: Go for applied math in an
engineering school. Start with a real
problem, hopefully a significant real
problem, likely from outside academics,
hopefully identified before or early in
the Ph.D. program. Do some new math to
get the first good or a much better
solution to the real problem. So, the
math is "new" -- got (i)! Since the work
is math, with theorems and proofs, it's
easy enough to check for "correct" -- got
(ii)! Since have the first good or much
better solution for the real problem which
is hopefully significant, get
"significant" -- got (iii). Note: The
math may not, just as stand alone pure
math, be seen as significant -- so, likely
have not proved the Riemann hypothesis,
shown that P = NP, etc. But compared with
a lot of pure math, are already ahead by
one point -- have one real world
application!

In my case, I started with a problem I had
identified in industry before grad school.
I found an intuitive and rough solution on
an airplane ride. In my first year in
grad school, one course I took let me make
solid math out of my intuitive solution; I
did that independently in my first summer,
walked out of the library with an 80 page
manuscript that had all the actual
research for my dissertation.

Then I encountered some of the nasty
nonsense as in the OP: There was a prof
who didn't like me. He thought of rows,
columns, and layers, lines, stay within
the lines, rules, etc. and resented that
I'd basically written my dissertation
independently within 12 months of arriving
on campus and before taking the qualifying
exams.

Well, a course had a question but no
answer. I did a good enough literature
search and saw that likely there was no
answer known -- since it was a very narrow
question, it was easy to do the literature
search. I got a reading course approved
to address the question and write a
paper, maybe just expository and maybe
without a solution. Just before getting
the reading course approved, in a few
evenings I found a rough solution. So,
got the course approved -- shook hands
with the prof. Then I cleaned up my first
solution, mostly sitting beside my wife on
our bed while she watched TV and I worked
on the problem, and found a much better
solution and a general result that was
surprising, even shocking, and settled
some related questions. That took two
weeks into the reading course. I
submitted my manuscript of about 20 pages,
and I was done with the reading course.
Fast course. The course also had three
credits and gave me the last credits I
needed for an MS. News of my work spread
through the department; my favorite prof
walked up to me in the hall, "I heard
about your result. It also says that
....". Yup, clearly it did.

The result was clearly publishable; later
I did publish it -- no problem, accepted
right away.

The biggie, practical result was that
suddenly I had a halo and a coat of Kevlar
armor against any criticism; any of the
faculty would have loved to have done what
I did. To defend against the abuse as in
the OP, I recommend -- do some publishable
RESEARCH.

(3) Handling Ph.D. Qualifying Exams. At
least at one time, the Web page of the
math department at Princeton stated, IIRC
(if I remember correctly):

"Students are expected to prepare for the
qualifying exams on their own. Graduate
courses are introductions to research by
experts in their fields. No courses are
given for preparation for the qualifying
exams."

In part, the qualifying exams are like a
foot race, but in this race you can get a
head start and be 1 foot from the finish
line when the starting gun goes off.

I consider that Princeton policy to be
somewhat wise. So, prepare for the
qualifying exams on your own (to be able
to do this in math, a good ugrad pure math
major should be sufficient to let you know
how to study and learn, make good
progress, and not get stuck or lost). To
do this preparation, get the best, focused
information you can on what will be asked.
So, get recommended texts, copies of old
exams, maybe chat with some profs and some
students who have taken, hopefully passed,
the exams, syllabi of any relevant
courses, etc., maybe all before applying
to the Ph.D. program. Then study. Use
some judgment on how deep to cut and how
many proofs to memorize -- cut deep enough
but not so deep you take too long or just
quit.

Then show up as a first year Ph.D.
student, do well in some courses, do well
on the qualifying exams, complete your
research, listen to Pomp and
Circumstance, get your degree, and LEAVE.

(4) Academic Career. If you want an
academic career, then maybe don't leave
the grad program so soon. Instead,
publish some papers, get some streams of
promising research going, meet people at
research seminars and conferences, get
known by the Editors in Chief of the
journals or conferences where you publish,
if invited to give a talk at a conference,
do so, do the usual meet and greet and
publicity, build your own professional
network, etc., hopefully some of your
profs will help you get some job
interviews, etc. Learn how the academic
games are played -- there are some really
important academic games, and you very
much should learn how they are played.
Then you will be in an academic career.

One possible prof slot is in a B-school.
Consider that. Someone with a good
applied math background has a heck of an
advantage in a B-school. Papers that make
progress on practical problems are
commonly considered good research in
B-schools. People who want to hire
consultants tend to regard B-school profs
as more practical , i.e., motivated by
money, than more pure profs.