Regarding the free shipping that is offered to Amazon Prime members - does that also apply to items sold by third-party sellers, or only to items sold by Amazon directly?

You can get Prime shipping if it's sold by third-parties but fulfilled by Amazon, but not all sellers do this.

It will always tell you during the order process. Normally it says "Prime" next to the item when you're shopping.

most (a definite majority) third party sellers do not honor prime shipping.

Yeah, this bugs me plenty. I've heard arguments on all sides but I still wish the shipping was just included as part of the price. Wish they would do the same for tax, also.
On the plus side as a Prime member, I have been taking advantage of the $1 digital credit for "no rush" shipping. There's little I cannot wait a few days for!