Yes, it is called the cofinality of the order. Just apply Zorn's lemma to the class of all well-ordered suborders of $P$, ordered by end-extension. If you've got a maximal such order, then it must be cofinal, since otherwise you could end-extend it. Note that Zorn's lemma applies, since the union of chain of end-extensions of well-orders is still a well-order.

Yes, it is called the cofinality of the order. Just apply Zorn's lemma to the class of all well-ordered suborders of $P$, ordered by end-extension. If you've got a maximal such order, then it must be cofinal, since otherwise you could end-extend it. Note that Zorn's lemma applies, since the union of chain of end-extensions of well-orders is still a well-order.

It would seem to be a weak choice principle to assert that every linear order has a cofinality, and I'm not sure exactly how this assertion relates to AC and its variations.

Meanwhile, in the case of partial orders, we cannot necessarily find such a cofinal subset, even when the partial order is upward directed. For example, the collection of all finite subsets of an uncountable set, ordered by inclusion. There can be no linearly ordered cofinal subset, since every node in the order has only finitely many predecessors.