By the way "failure" means the guaranteed data rentention time is not guaranteed anymore. It does not mean that it stops working immediately. It means that it may lose memory sooner. That is: if you do not need a high retention time, you can overwrite it much more often degrading the retention time further and furter.
Is it each bit that is written count as one (so a byte would = 8) or is it each time the function is called?
If I code something to write and I make a mistake so that it loops will It quickly use up the 100,000 limit?
I've been told that EEPROM can work fine, through many write cycles, until the power is turned off.
Yes this is true. Back in 2001 ATMEL were selling a flash part that they claimed to have a endurance of 1000000 write cycles. I was involved in testing them and found the real life was only 80 to 120 cycles if you powered then down between writes. That was flash not eeprom but I imaging the same applies.