If a creature enters the battlefield with counters under another player's control, that player is the player who puts those counters on it, even if you control the spell or ability putting that creature onto the battlefield.

Support can target a creature another player controls.

Generous Patron's last ability triggers even if the counters you put on another player's creatures aren't +1/+1 counters.

You can't put more than one +1/+1 counter on any one target using the support action.

If a spell with support has other abilities that target creatures, those abilities and the support ability can target the same creature.

If you put one or more counters on multiple creatures you don't control at the same time, such as by supporting two different creatures you don't control, Generous Patron's last ability triggers for each of those creatures.

If some, but not all, targets for a spell become illegal, the remaining targets are affected as appropriate. If all of a spell's targets become illegal, that spell doesn't resolve.