Everyone in NYC knows the subway always runs, so I expect the way
people use it is different from cities where late night service is a
recent or sometime thing.

Is it safe in all parts of the network at 3am?

As safe as it is at other times of the day. By European standards NYC
is somewhat dangerous because we have so many guns floating around
(although nothing like Florida or Texas), but by US standards it's
quite safe.

I have read that in general subway stations are safer than the street
above or below.