Remember NH is an open primary and democrats can vote for Haley
Yes but traditionally this does not happen as much as true 'open' primary states because in NH (and a few other states do this too) a Democrat who chooses a Republican primary ballot becomes registered to the Republican Party.
Additionally, Biden unwisely choose to stay off the ballot in NH in deference to the Dems gerrymandering their own primary order, now there is a fairly serious effort to write him in. That alone will keep crossovers under previous election years.
False.
The primary isn’t open. Rather, if you are a registered Democrat, you can only vote in the Democrat primary. If you are unaffiliated, you can vote in either the Republican or Democrat primaries.