You can count people - three people, 20 people, 135 people, etc. so it is not a mass noun, like "sugar" or "anger."
Therefore, use "are": There are a lot of people.
(There is a lot of sugar in this pie. There is a lot of anger over this proposal.)
Yes, but I think the difficulty is that there is only one 'lot', no matter how many people there are in it.
How about, "There is/are a houseful of people down the block who I don't like". (I'd say 'is')
I say "There's a lot of people", but not "There is a lot of people." The contraction "there's" works for plurals.
Is your question purely about "a lot of ..."?
I would also say, "There is a roomful of people ...", so it depends on what X is in "There is/are an X of people."
I don't think there is a final conclusion.