Hello Anon
For #1, I would myself use a comma, unless I intended a particularly abrupt or tart tone:
1. No gifts please!
For #2, I'd say it depended on the intonation:
2a. The performance is intended for adults, but children are invited too.
— no pause before "too".
2b. "That's very strange. Have you seen the note from Esmé about the party on Thursday night? Children are invited, too." "Well, count me out, then."
— pause and downward intonation, before "too".
That said, I'm sure other members will disagree with me, so you may still get the advantage of those English majors. (Though sometimes there's no arguing with military types, especially English ones.)
MrP