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 use the All Turns effect on BT17-087 Marcus Damon to gain memory if it's treated as a Digimon and suspends while my opponent has "your opponent can't gain memory other than by Tamer effects" activated?

Yes, you can gain memory. 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.