I can’t comment on any official teachings, but I can tell what I think…
1.) There are many times in the bible when God grants someone a new name to indicate His new relationship with the person. For instance, Abram–> Abraham, Simon–>Peter (The Apostle), Saul–>Paul. The name change reflected the spiritual change in the person when God made them new in some way.
When a person receives the Sacrament of Confirmation (and also baptism), God also makes us new, or confers upon us new graces. The new name we choose is a reflection of this. The idea is that by choosing to name ourselves for this person we are also asking whatever saint it is to intercede for us, and we are possibly hoping to model our own lives after some exemplary part of that saint’s life.
For example, when I was confirmed, I chose the confirmation name Therese, after St. Therese the Little Flower. I chose her specifically because I thought she was a neat person and her childlike way seemed so unlike mine (I can be very pig-headed and obstinate). I wanted to become a bit more childlike, since this is what Christ asks us to be toward him
It’s been 10 years since my confirmation, and I’m still working on that.
2 ) As far as whose names are available, I think pretty much everyone who is officially canonized by the Roman Catholic Church.
3.) I don’t think Constantine is among those, but I could be mistaken. There are
a lot of saints out there.