Answer:

1. yes, as long as all the title holders sign the deed transfer document. Actually this has been done pretty often to avoid probate. That answers part of question 2 as well.
2. Whether the spouse will elective share or not depends on state law. As you know, they vary from state to state. If it is in New York, the primary residence is still part of the estate for elective share basis. Basically there is almost no way to disinherit your spouse in New York using a trust. There is only one complication: if you hu*****and bought the house before your marriage, this can get complicated. Also I am not sure if you have any pre-nup agreement. As for your hu*****and's claim, I don't think it would be too difficult to add your name on to the deed. But maybe in your state it requires a strawman to complete such a transaction, meaning you have to find a third person to take the title from your hu*****and and transfers it right back to you and your hu*****and jointly.