When an ability triggers, all the game does it make a note of it. The triggered ability is placed on the stack the next time a player would get priority. This is only after Hypergenesis has finished resolving.

603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.

603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. [...]

So 1) yes, and 2) no.

When an ability triggers, all the game does is make a note of it. The triggered ability is only placed on the stack the next time a player would get priority. This is only after Hypergenesis has finished resolving.

603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.

603.3. Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. [...]

So 1) yes, and 2) no.