I'm sorry to hear about the problems you're dealing with right now, but I am glad to hear that you've decided to start taking your faith more seriously.
I am shocked that your wife would tell you "that she will seek intimacy elsewhere" if you don't sleep with her again soon. That's just beyond horrible. I understand that in a marriage we have duties to our spouses, but it is not as if you simply stated that from now until you die you will never be intimate with her again. I really hope your wife was not serious about her comment, because if she was that would be a big indication to me that her love for you isn't very strong. I mean what if you were to get into an accident and no longer be able to be intimate afterward? Would your wife cheat on you or leave you because of it? If not, then why this radical reaction over something that is only going to be temporary once you can have your marriage validated?
I think you, your wife, and your priest all need to sit down and talk together about this. You all need to be on the same page.
As far as your wife's tubal ligation goes, from what I understand (and I could be wrong) if your wife sincerely goes to confession and receives absolution for this you should be ok. It's something she had done in the past that she most likely could not have undone. So as long as she were to acknowledge the sinfulness of her action and repent then I don't see this being a barrier to having your marriage validated in the Church, even though your wife's operation makes it impossible for your marriage to be fruitful now.
I do think one thing you'll definitely have to work on though is your wife's understanding of the faith. If this is going to work she's going to need to better understand where you and the Church are coming from regarding the various issues she has with Church teaching. Again, this is a reason for you to both sit down with the priest and discuss the situation.
Anyway, I wish I could give you more and better advice. To be fair though, I should let you know that I have never been married and am only in my late 20's. So my understanding of how things should be done in this situation is based off of limited and incomplete knowledge.