Any tied note must be the same as the note it's tied to. Otherwise it's a slur!. So an accidental sharp on a G will make it G♯, and the note it's tied to will also have to be a G♯.
Heather is correct, assuming the G note (tied) is in the next bar. At that point, the bar line has officially cancelled that accidental - except it can't for the tied note! One rule breaks another! So if there is another G♯ needed in that following bar, it'll need another accidental anyway - not even a courtesy/advisory one, but one in its own right.
(EDIT): Further to the question, now with example. Since the second G♯ has a sharp attatched, which strictly speaking isn't necessary, it does mean that the last G of the triplet must also be G♯. I'd have put a ♯ sign in brackets just before it - making it crystal clear.
On your second point, if the publisher/writer is good enough to put an advisory sharp for the tied note, surely they would also be good enough to put in an advisory natural sig if the next G, even though in the next bar, needed to be natural. Otherwise leave out all the advisories!