A lot of nice stuff goes "wrong", or at least different in characteristic 2. Perhaps the most interesting possibility remaining is that the unit conjecture is true for fields of characteristic != 2 but false for 2?
Indeed, two is the oddest prime
I wonder if there are any published proofs or theorems that, directly or indirectly, relied on this conjecture being true.
Multiplication was the hardest, and several numbers were hardly invertible. OpenSSL has a nice inverter. Could be used to make crypto better.
Or possibly not. I'm reminded of Sarah Flannery's idea that matrix multiplication might be a good one-way function for crypto. Turned out that in that context, there was a cheap inversion that wasn't immediately obvious. Her memoir/math book about this (and learning mathematical cryptography) co-written with her father is a good introduction to a lot of mathematical cryptography ideas: http://www.amazon.com/exec/obidos/ASIN/1565123778/donhosek
Inversible hash functions (such as murmur) are only related to cryptographically strong hash functions in that they share the words "hash function".
The article is about groups of units in group rings. That seems to have very little to do with your post - am I missing something?