Must something be unprovable to permit faith in it? Is it possible to have faith in something which is certain provable fact? If something is proven beyond any doubt can you still have faith in it? How?
The much used ode to faith in the NT book of Hebrews tells us a little bit about what faith is. Here are the first couple verses:
The passage goes on to give many examples, but this bit helps us scope out a definition of faith that is somewhere between the two extremes you suggest in your question. First of all we see that there is assurance. True faith is not blind, it is based on evidence and knowledge. It is a kind of trust. Trust between two parties is established in past encounters but looks forward to the future knowing what to expect from the other party. Secondly it is a conviction. This doesn't have to be without any doubt (in fact many of the people listed as examples in the passage experienced doubt) but it does have to come to a conclusion and really believe/trust.
Faith is built on evidence and knowledge as well as a trust that enables us to be sure the parts we don't fully understand will work out.
Faith and proof are not mutually exclusive -- faith projects evidence of past events into the future.
Something need not be unprovable for a person to have faith in it. For one thing, something might be provable but a person may not have access to the evidence. For example, if you flip a coin, I may have faith that it came up heads. This is certainly not unprovable -- someone can look at the coin. But if I haven't looked at the coin, then I can certainly have faith that it came up heads.
But even if something is provably true and a person has access to that proof, he could still believe the claim on the basis of faith as an independent source of belief. That is, the faith itself is not based on proof, but there is no reason the two sources of belief could not exist concurrently.
For example, if I have faith that a coin came up heads and then I look at it and see that it came up heads, nothing stops me from continuing to have that faith as well. Thus, if I found some error or defect in the proof, the belief would remain, sustained only by the faith.
When we say faith is "in the absence of evidence", we mean that evidence is not required for faith. But "independent of evidence" would perhaps be a more accurate term.
One can argue there is no need for faith if one has evidence. And it is true there is no need for faith to have the belief if one has evidence. But sometimes having the belief is not the only thing that's important. Sometimes the route is as important as the destination.
The New Testament people of Jesus' day did not think faith and proof were mutually exclusive. They used reason to lead them to faith or against it:
The people were not believing based on blind faith, but were comparing Jesus and His works against what the OT had to say about the Christ. God gave specific prophecies about the Messiah by which they could judge whether they should believe that He was the Christ or not. In other words, they used reason to seek proofs before faith.
Look at their reasoning: "When the Christ comes, will He do more signs than these which this Man has done?" That's pretty reasonable, since the Messiah will do certain kingdom-oriented miracles when He comes, and the Messiah could not do more or better miracles than were done by the hand of Jesus.
Also, "Will the Christ come out of Galilee? Has not the Scripture said that the Christ comes from the seed of David and from the town of Bethlehem, where David was?" Once again, they judged His life by the Scripture, not blind trust (faith is trust; the element of proofs is not part of its definition any more than "believe" is defined by proofs). They did not know that He was indeed born in Bethlehem (southern Israel),]1 for He had been living in Nazareth and Capernaum (northern Israel -- [Matt. 4:13) after His family moved back from Egypt.
It depends what you mean by "faith".
When regarding religious matters, the most typical definition of "faith", used by religious and non-religious alike--and I'd argue this is the most useful definition in these cases--is: belief in the absence of evidence (and I mean the post-Enlightenment understanding of evidence: independently confirmed by high quality sources, objective, fits into a successful model of the way nature works, reproducible or repeatedly observed, stands up to testing, etc.)
Under this definition, then, the answers to your questions would be:
Either unprovable or as yet unproved.
With this definition, no; then it is no longer an act of holding faith, it is an act of knowing.
No, not with this definition.
You are buying into atheist ideology if you think that there is no proof or evidence for God's existence
No I do not think so. When New Testament scholars decide on the validity of Jesus' resurrection they do so on the basis of historical facts.
What are these facts I can imagine someone ask?
Now we are left with an event that needs explaining.
How did the tomb become empty?
Many New Testament scholars posit then the idea that his bodily resurrection was the best explanation for this empty tomb. Now not everyone might believe this and it does not prove the resurrection to everyone, but it definitely gives sound and rational basis for a belief in Christ's divinity.
We are very privileged as Christians to have a religion that makes historical claims that are not ambiguous by nature but can be verified on a historical basis.
So in closing. No I do not think faith and proof are mutually exclusive. I think you need proof to have any conviction in the miraculous and I think we have that.