I am in middle of a battle with former leasing management for my deposit in TEXAS. We already went to Texas JP Small claims court. I won. Now the other party appealed it to County Court level. They are trying to scare us with a lawyer and questioning and so forth. Several questions

1. Who is the Plaintiff and who is the Defendant in the new case?

2. Can the lawyer force us to drive from a different state for a deposition? (another way of costing us money)

An appeal is based on the original case, so the roles stay the same. I would check with the court. Normally a deposition is held at the deposed's convenience.

... Can the lawyer force us to drive from a different state for a deposition?

Very much doubt it. Check the civil rules for the court the appeal is being heard in. If they don't say, I would move for a protective order.

And I question whether they can institute discovery, including depositions, in an appeal.

Generally an appeal is based on a procedural flaw by the original court or the discovery of new evidence. You can't file an appeal based solely on not agreeing with the court ruling.