idioms - Is there a name for this kind of phrase?

I believe this is a Chiasmus:

Chiasmus is a figure of speech where wit is conveyed through the reversal of words or phrases in clauses.

Some examples would be:

'Never let a fool kiss you, or a kiss fool you'
"Nice to see you, to see you, nice!"
"You forget what you want to remember, and you remember what you want to forget."

That's for your first example.