Sure there is a libertarian influence. There is no official libertarian Democratic congressional group such as the CPC, the Blue Dogs, and the NDC, but there is the Democratic Freedom Caucus. Ever heard of Mike Gravel? He is a Democrat but he sought the Libertarian Party's nomination for President in 2008.
Nope. There are many pro-life Democrats. Check out Democrats for Life. That group claims 1 in 3 Democrats are pro-life. Of course the national platform is pro-choice but that doesn't mean there aren't pro-life Democrats. Just like all Republicans are not pro-life.
The Clintons are not progressive Democrats. They came to power during the second Third Way movement and Clinton built his power base through the Democratic Leadership Council. There is a reason Sanders challenged Clinton for the nomination.
Obama was more of a progressive than the Clintons, I will give you that. But still, Obama was more aligned with the New Democrats than he was with the progressive wing of the party.
If a true progressive was elected President, those on the right would quickly see the differences between that progressive and the Clintons.
No worries. This is just a message board, not an English class.