Question 1: Does there exist models of the ZermeloFraenkel set theory without the axiom of choice, but such that every indexed family of nonvoid sets whose index set has a wellorderable cardinal admits a choice function ? Question 2: The same as question 1, with "a wellorderable cardinal" replaced by "a linearly ordered cardinal". Gérard Lang
For the first question: Yes. It is known that the axiom "Every wellorderable family of nonempty set has a choice function" implies $DC$ but not $DC_{\aleph_1}$. You can find the proofs in Jech's "The Axiom of Choice" and in Felgner's "Models of ZFSet Theory". I am not sure about the second question, but I believe the answer should be "yes" as well. In Consequences of the Axiom of Choice you can find the following principles:
Entering those three numbers in the table show that $1\implies 202\implies 40$ and neither implications is reversible. Although in the case of $202\implies 1$ this is in a weaker axioms system without regularity. Edit: I had a bit more time now, so I went to chase after the cited source for Form 202:
Here is an excerpt from the review:
So it seems that in ZF, requiring for linearly ordered families is enough to require the full axiom of choice. 

