Hey, could you please tell me if I need to install any patches for the Heroes of Might & Magic V Gold edition? And if so, which ones? Thanks!

The only relevant patch for the Gold edition is patch 3.1 for Tribes of the East. Check the website's file archive.

Hey, can anyone tell me how to deal with the AI slowdown in long missions?
It's just unbearable to play for 3-5 months. The computer takes 10 minutes to make a move. What is it doing?
Is there any way to reduce its turn time?

That's a characteristic of the AI in the fifth installment, and I haven't heard of any reliable ways to fix it. But 10 minutes seems excessive; with reasonably modern processors, you should have to wait less. The only thing I can suggest is not to set the maximum number of computer players on the map; 3-5 might be quite enough even on large maps.