Yes, when the total bitcoins will be mine the transaction cost will definitely increase as it will be only the source of income for the miners at that moment. Though I can’t surely say this as the world is moving very fast wrt the innovations.
I doubt that. Transaction cost may go cheaper and faster. Many things could happen.
One of them is some miners giving up because they think it will not be profitable anymore. The difficulty level will go lower and each block will be easier to find.
The other thing is new computing powers. We don't know what will be invented at those times since we will be dead before the bitcoin mined is depleted.
We are humans, we will keep on growing. Upgrading.