Are you able to sign a player during the end of year week before the league roles over?

Yes

The one danger here is if a player doesn't immediately re-sign, which happens sometimes (but not often). If this happens on the EOS sim, then the player becomes a FA.

Also, if you're trying to sign a free agent before EOS spin, you'll have to pay for the current year even though he'll never play a down until next season.

It didn’t sign him at the spin.

You can unless the contract will take you over the salary cap, in which case the offer will automatically be rejected.