Let's tackle your second question first. Yes, leafless poison ivy still contains Urushiol, and still can give you a rash. Here are three sources supporting this statement: a first-aid sales company, a newspaper, and a non-peer reviewed infosheet from a university. Be careful with poison ivy all year.
Your first question has slightly more nuanced implications. To know if there is a time of year where above-ground poison ivy doesn't contain Urushiol, first we have to think about when and where the urushiol is produced in the plant cell, where it is stored, and how it degrades. Some studies have shown that Urushiol and like compounds are present throughout the plant. In this older article, they extract it from both the leaves and the twigs, but more current advances have been able to focus on urushiol in just the stem. I have seen no evidence that it reduces at any time of year, and very little publication on how it is produced biochemically, although people are working and have worked on the topic. I am interested to learn more about this aspect from other answers to this question.
(An aside: you can also pick up a Urushiol rash from some cheap furniture, so this compound can stay poisonous on a wood surface for quite a while. Unfortunately, I'm speaking from first-hand experience, which is backed up by this informative article about Urushiol!)