Okay.
Nothing overly specific. The X-Men are mentioned as an active mutant group and some of the kids wonder if they'll be transferred to them when they complete Dr. Reyes' program, but that's it.
The Institute is not mentioned period.
Yep, there is. Spoil at your own risk: spoilers:end of spoilers
When Dani is being tested during the second act, she gets a telepathic flash of Dr. Reyes' mind, of mutant children being forced into combat training. Stock footage from the Logan is used to represent it (it's a bit from the nurse's cell phone recording of Transigen, where a boy is training against dummies but then starts attacking the guards). While Dani and the other New Mutants only work out that Reyes and her unseen superior have been secretly planing to weaponize them, not help them -- the audience learns that Reyes is in the employ of the Essex Corporation (from Apocalypse).
Despite the stock footage, there's no further evidence to prove or disprove of Transigen from Logan is a subsidiary or future incarnation of the Essex Corporation or a separate organization (in other words, the situation is exactly the same as before). The New Mutants do make it clear that mutants are relatively common and the New Mutants themselves are too young to belong to the last generation described in Logan, directly contradicting that movie's premise that mutants are all but extinct and are no longer being born (but that's business as usual for the series.