Rambam, Hilchos Melachim 3:2:
ואם הוסיף אחת ובעלה לוקה
If he adds an extra wife and has relations with her, he gets lashed.
So that answers question #3. As for the others:
On the face of it, kiddushin ought to take effect like in any other case of חייבי לאוין (according to everyone except R' Akiva, and maybe even according to him, since the issur isn't really on marrying her as an individual, just as wife #19). Although Minchas Chinuch 501 understands the Chinuch to be saying that the issur is violated already when the kiddushin is done, and in that case that maybe it doesn't take effect, under the rule כל דאמר רחמנא לא תעביד אי עביד לא מהני.
Would he have to divorce her? Probably, since otherwise he's continuing to do the aveirah. Minchas Chinuch there says she'd need a get because of the safek above, whether the kiddushin took effect.
In a comment on the OP, @heshy asked whether she'd become a chalalah (or rather, a zonah). Minchas Chinuch there says no, because it's not an איסור השוה בכל.