I fly Delta regularly. We took our, at the time, 8 month old on a trip to Chicago. Out of 4 segements, they let us pre board on 3. On the other one, we were told it was Delta's policy that they no longer allow preboarding of people with small children.
The ironic part is the reason she gave. She said it was because they typically tend to clog the aisles with all their stuff. I didn't point out that would be a reason FOR preboarding.

It is not a huge deal, depending on where you are flying. Most people are fine and in many cases helpful.

As far as the stroller, you will likely have to gate check it, but usually by the time you get off the plane, it will be planeside waiting for you.