As others have said: Yes, it does count. You’re still making a formal, legally-binding pledge, in front of witnesses, to tell the truth, the whole truth, and nothing but the truth.

By the way, you don’t actually need to swear at all. If you’re an atheist, you can instead make a solemn affirmation, which is the same thing but without invoking a deity.

Historian Brett Devereaux goes into the finer details of oaths here. The idea of invoking the names of deities when one makes a promise is as old as religion. The idea is that doing so calls the attention of that deity, who then acts as a witness and enforcer of the promise (so make sure it’s carefully worded!) Thus, in ye olden dayes, if you swear by God to tell the truth and then tell a lie, the punishment is damnation in Hell for breaking a promise that God formally witnessed. Touching a Bible (or a relic or religious artifact) makes the oath more potent.

Of course. If you promise to do something, but don’t intend to follow through, did you still make that promise? Yes you did. If that promise was legally binding (as in an oath of office or oath before a court) you can of course be held accountable for that oath whether you “believe” in the thing or not. Can you imagine what a mess it would be if you could just say “well I don’t believe in the Bible, so I can lie in front of a judge all I want since they made me swear on a Bible”?

The whole point of swearing on the bible was to get God to be involved in enforcing the oath. So your disbelief may negate this aspect. On the gripping hand, you also swear in front of a government representative to get the gubmint involved in enforcing the oath. And if it is in their interest to enforce the oath; they will. With guns if necessary.

yeah, because the invisible sky father will smite you irregardless of your beliefs