How are safety/liveness languages defined on the set of finite or infinite words?
The answer is yes to all questions, so it is enough to answer 2 and 4, as the definitions work in particular for languages of finite words:



A language $P\subseteq \Sigma^*\cup\Sigma^\omega$ is safety if whenever $u\notin P$, then $u$ has a finite prefix $u'\in\Sigma^*$ such that for any word $v$, $u'v\notin P$.


A language $P\subseteq \Sigma^*\cup\Sigma^\omega$ is liveness if for any finite word $u\in\Sigma^*$, there is a word $v$ such that $uv\in P$.