If I make this ahead of time and keep it in the fridge, should I cover the top with a damp paper towel to keep the phyllo fresh?

In working with phyllo, I wouldn't keep the paper towel on it, as it can make it wet and ruin the composition (make it soggy and break). I've done cheese pies the night before I just butter between the layers and butter the top, and they cook up fine the next day.

Yes you can...or bake it ahead of time and re warm in the oven with a little more melted butter brushed on the top. Actually you can make this days ahead and freeze without the melted butter. Thaw it a bit, add the butter and bake as usual.

I would! And maybe even plastic wrap over that