Q1: What does the "your opponent can't gain memory other than by Tamer effects" effect do, exactly?

With this effect, your opponent's memory will no longer increase by any effects except Tamer effects. This effect affects the opponent player.

Q2: Can I gain memory using an effect on a Tamer that's treated as a Digimon with X000 DP due to an effect while my opponent has "your opponent can't gain memory other than by Tamer effects" activated?

Yes, you can. When an effect is activated on a card that's treated as both a Digimon and a Tamer, the effect is treated as both a Digimon effect and a Tamer effect, therefore you can gain memory.