I’ve read it can take up to a year or more to process an annulment.
This depends on the diocese and how many cases they have. It also depends upon the complexity of the individual case, the documentation provided, etc.
It can range between a few months and over 1 year.
In that time period, can we get engaged?
It is not prudent to get engaged, which typically starts a whirlwind of well-wishers, party planning, wedding planning, and expectation setting. If you were to get engaged and then find out he was not free to marry, how hard would that be? Who would you “let down” other than just him? Parents? Friends? Family?
Honestly, engagement should not come until you know he is free to get married.
But, it is something you should talk with your priest about.
Can we start the premarital counseling before the annulment is finalized?
That is an excellent idea.
So if I get engaged to today, it will be a minimum of about 2 years before we can get married?
Yes, it could be that long, or longer. And, engagement leads to a set of expectations and behaviors you probably want to avoid until his situation is more clear.
At least talk to your priest first.
Also he has not had any contact with the ex, so how is he supposed to provide her whereabouts?
He can provide the tribunal with what info he has. He is not required to track her down. The marriage certificate, divorce decree, and last known address are all he would need to give them.
And, I see you started to get angry and asked questions about how the Tribunal has authority in the matter-- again I really encourage you to get the book and read it as this is also covered. But, in short, Christ himself established the Church and gave the Church-- through the Bishops-- to bind and loose. See Matthew 16:15-20 and Matthew 18:18. Binding and loosing includes making determinations of this kind.