What about after that point in time, does DJT have to make ongoing interest deposits as the appeal drags on?

I don't know, and a quick google search didn't get into that level of detail. The actual rule doesn't address that, so if you just read the rule you get the stay when you post the bond, and it doesn't say anything about having to top it off in order to keep the stay in effect.

Will the State of NY pay interest to DJT on the amount of the bond over and above the final judgement?

Not if it's a bond, no, because the actual assets aren't deposited with the court. IOW, when you get a bond, you get a surety company to commit to the court that they will pay the amount if you fail to. Typically, you'll pledge collateral to the surety to get that bond. But since you retain ownership of the collateral, you keep it in your own bank or brokerage account - so you'd earn interest over time. The court would just partially release the bond, so that it matches the revised judgment.

I don't know what would happen if the appellant deposited cash with the court itself - presumably that gets put in an interest bearing account, but I don't know how the interest is allocated.