If I play Erika's Invitation and put a Basic Pokemon onto my opponent's Bench would they put two damage counters on that Basic if I have Gapejaw Bog in play?
Yes, since the Pokemon is coming from their hand onto the Bench Gapejaw Bog places 2 damage counters onto it.
Source: TPCi Rules Team (2023-11-09)
When using Erika's Invitation can I choose to NOT put any Basic Pokemon onto my opponent's bench, or do I have to put at least one down if I'm able to?
If you find a Basic Pokemon in your opponent's hand you must put it onto their Bench.
Source: TPCi Rules Team (2023-09-28)
Can you play Erika's Invitation if your opponent's Bench is full?
No. It's public knowledge that you can't put down any additional Basic Pokemon on your opponent's Bench, so you cannot play the card.
Source: TPCi Rules Team (2023-09-28)