Issue 1:
Kind of an edge case I would say, but still it would be nice to get an error or have an automatic renaming of the columns. If a simple error is acceptable as a first step I could definitely put together a PR.

The following snippet shows that when you concatenate a column of zeros and a column of ones the resulting dataframe has two columns of ones:

import pandas as pd
import numpy as np
ts0 = pd.Series(np.zeros(5))
ts1 = pd.Series(np.ones(5))
ts0.name = ts1.name = 'same name'
df = pd.concat([ts0, ts1], axis=1)
print df

and the output (pandas 0.10.1):

same name same name
0 1 1
1 1 1
2 1 1
3 1 1
4 1 1

whereas I would expect:

same name same name
0 0 1
1 0 1
2 0 1
3 0 1
4 0 1

Looks like this is due to pandas.tools.merge._Concatenator._get_concat_axis which doesn't check for name uniqueness, here is the relevant snippet:

names = []
for x in self.objs:
if x.name is not None:
names.append(x.name)
else:
return Index(np.arange(len(self.objs))
return Index(names)

Issue 2:
Code Sample, a copy-pastable example if possible

# Note: This is not real Python code. I can't reproduce these problems with toy data, but they
# show up consistently with certain sets of real data.

from pandas import Series
vocabulary = Series(data = <a few thousand integers>, index = <a few thousand words/tokens, including u"null">)
new_vocab = Series(data = <a smaller list of integers>, index = <a smaller list of words, mostly overlapping with the first one, also including u"null">)
new_words = new_vocab.index.difference(vocabulary.index)

print u"null" is in vocabulary.index
print u"null" is in new_vocab.index
print u"null" is in new_words

> True
> True
> True

from pandas import DataFrame
matrix1 = DataFrame(data = <a bunch of numbers>, index = <list of topic names>, columns <list of words/tokens, _not_ including NaN or u"nan">)
matrix2 = DataFrame(data = <a bunch of numbers>, index = <list of topic names>, columns <list of words/tokens, _not_ including NaN or u"nan">)
matrix3 = matrix1.multiply(matrix2)

from numpy import NaN
print NaN in matrix1.columns
print u"nan" in matrix1.columns
print NaN in matrix2.columns
print u"nan" in matrix2.columns
print NaN in matrix3.columns
print u"nan" in matrix3.columns

> False
> False
> False
> False
> True
> True

Problem description

This may be related to #18988.

I feel bad about reporting these bugs, because I can't duplicate them outside actual scripts with actual (rather voluminous) data. The first common denominator to all the issues is that when I perform operations involving two objects, I end up with extra columns (in DataFrames) or elements (in Series or Indexes): sometimes duplicates of existing columns (but the second, duplicate column will contain all null values), sometimes entirely new columns/elements.

The second common denominator is that every case seems to involve "null-like" index values: NaN, u"nan", u"null", u"n/a", and u"null.1.1" are the ones I've actually observed. In one case, u"null" creates a problem; in another u"null" works just fine, while u"n/a" and u"null.1.1" create problems (but u"null.1" doesn't). In a third case, the null-like values don't exist in the original objects, but show up after an operation, as new columns (interestingly, in this case, under pandas 20.3 I got two columns, NaN, and u"nan", while in pandas 22.0, I get those two plus u"n/a"--or actually, the "len" function is telling me there are four extra columns, but I using a "difference" method only gives me a list of three)--and they show up with some sets of data but not others.

In each case where errors have shown up, I've added code to check for them and remove the spurious columns/elements, but it's been very frustrating to run down different problem that can occur.

Output of pd.show_versions()

[paste the output of pd.show_versions() here below this line]

INSTALLED VERSIONS

commit: None

python: 2.7.14.final.0

python-bits: 64

OS: Windows

OS-release: 10

machine: AMD64

processor: Intel64 Family 6 Model 94 Stepping 3, GenuineIntel

byteorder: little

LC_ALL: None

LANG: en

LOCALE: None.None

pandas: 0.20.3

pytest: 3.2.1

pip: 9.0.1

setuptools: 38.4.0

Cython: 0.26.1

numpy: 1.13.3

scipy: 1.0.0

xarray: None

IPython: 5.4.1

sphinx: 1.6.3

patsy: 0.4.1

dateutil: 2.6.1

pytz: 2017.2

blosc: None

bottleneck: 1.2.1

tables: 3.4.2

numexpr: 2.6.4

feather: None

matplotlib: 2.1.0

openpyxl: 2.4.8

xlrd: 1.1.0

xlwt: 1.3.0

xlsxwriter: 1.0.2

lxml: 4.1.0

bs4: 4.6.0

html5lib: 1.0.1

sqlalchemy: 1.1.13

pymysql: None

psycopg2: None

jinja2: 2.9.6

s3fs: None

pandas_gbq: None

pandas_datareader: None