Obama was elected on hope and change, and the only specific policy proposal he had that made him different from Clinton was “no individual mandate”.
Yes, you’re exactly right in your thinking.
For the party out the presidency, the party chair is quite important for establishing priorities, building consensus, setting strategy, fundraising, etc. This is why it tends to be someone that everyone in the party like and respects, and is ideologically orthodox within the party. And the job is especially important for the DNC, given their poor position and given that many of their major party figures – are in their late 60s or 70s, and the rising generation hasn’t really been identified yet. The fact that the two leading candidates for the DNC are both very progressive tells you something about where the party is likely to go in the near future.
For the party with the presidency (especially in the first-term), they’re doing the same job, but more or less under the direction of the president.