As stated in the initial bug description, this is absolutely a fresh install: "Uninstalled Inkscape entirely (made sure "C:\Program Files\Inkscape\" does not exist)" I did this as a first step since I understand a corrupt installation is a likely culprit and did not want to waste your time.
There *may* be some incompleteness going on as I did not see any "import hashlib" line in the command output.
I ran it -v -v and got this:

...
import atexit # precompiled from C:\Program Files\Inkscape\lib\python2.7/atexit.pyc
ERROR:root:code for hash md5 was not found.
Traceback (most recent call last):
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 147, in <module>
    globals()[__func_name] = __get_hash(__func_name)
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 97, in __get_builtin_constructor
    raise ValueError('unsupported hash type ' + name)
ValueError: unsupported hash type md5
# trying _sha.pyd
# trying _sha.py
# trying _sha.pyw
# trying _sha.pyc
# trying C:\Program Files\Inkscape\lib\python2.7/_sha.pyd
# trying C:\Program Files\Inkscape\lib\python2.7/_sha.py
# trying C:\Program Files\Inkscape\lib\python2.7/_sha.pyw
# trying C:\Program Files\Inkscape\lib\python2.7/_sha.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_sha.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_sha.py
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_sha.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_sha.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_sha.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_sha.py
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_sha.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_sha.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/_sha.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/_sha.py
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/_sha.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/_sha.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\site-packages/_sha.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\site-packages/_sha.py
# trying C:\Program Files\Inkscape\lib\python2.7\site-packages/_sha.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\site-packages/_sha.pyc
ERROR:root:code for hash sha1 was not found.
Traceback (most recent call last):
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 147, in <module>
    globals()[__func_name] = __get_hash(__func_name)
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 97, in __get_builtin_constructor
    raise ValueError('unsupported hash type ' + name)
ValueError: unsupported hash type sha1
# trying _sha256.pyd
<snip a lot of repeated errors for sha256 and sha512>
# trying C:\Program Files\Inkscape\lib\python2.7\site-packages/_sha512.pyc
ERROR:root:code for hash sha512 was not found.
Traceback (most recent call last):
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 147, in <module>
    globals()[__func_name] = __get_hash(__func_name)
  File "C:\Program Files\Inkscape\lib\python2.7/hashlib.py", line 97, in __get_builtin_constructor
    raise ValueError('unsupported hash type ' + name)
ValueError: unsupported hash type sha512
# trying _hashlib.pyd
# trying _hashlib.py
# trying _hashlib.pyw
# trying _hashlib.pyc
# trying C:\Program Files\Inkscape\lib\python2.7/_hashlib.pyd
# trying C:\Program Files\Inkscape\lib\python2.7/_hashlib.py
# trying C:\Program Files\Inkscape\lib\python2.7/_hashlib.pyw
# trying C:\Program Files\Inkscape\lib\python2.7/_hashlib.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_hashlib.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_hashlib.py
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_hashlib.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\plat-win32/_hashlib.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_hashlib.pyd
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_hashlib.py
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_hashlib.pyw
# trying C:\Program Files\Inkscape\lib\python2.7\lib-tk/_hashlib.pyc
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/_hashlib.pyd *******
# trying binascii.pyd
# trying binascii.py
# trying binascii.pyw
# trying binascii.pyc
<snip binascii attempts>
# trying C:\Program Files\Inkscape\lib\python2.7\lib-dynload/binascii.pyd
import binascii # dynamically loaded from C:\Program Files\Inkscape\lib\python2.7\lib-dynload/binascii.pyd
# trying struct.pyd
...

It loads atexit OK, shows details about the hashes, then goes on to binascii

Yes, \lib\python2.7\lib-dynload\_hashlib.pyd does exist: see the ******* line above. Is something being loaded out of order?