Is "a lot" an adjective?
No. You may be confusing "a lot" with "a lot of". The "of" changes the part of speech.
"A lot" means "to a large extent or degree", and it is a countable concept. It can appear to describe the number of objects, the extent of an action's impact, etc.
I understand that it can be confused to be an adjective, since "a lot of" is an adjective. It could be a common mistake, but "a lot" (independent of the "of", but "of" may still be next to it) can be an adverb, or it could be an noun.
NOUN: "A LOT"
- He had a lot of time to tweak the settings to ensure that he perfected the programming---he was so into it that he lost track of time and finished after midnight.
One might replace "a lot" with "a boatload" or "an obscene amount", but not "much"; the time wasn't exactly countable, since in this situation, the person wasn't paying attention to the time (he lost track of time).
Then, what we have is that "time" qualifies "a boatload". He had a boatload of what? Time. Therefore, "time" is the adjective but "a lot" is the noun.
ADVERB: "A LOT"
- He worked a lot last night to get his finishing touches done.
"Worked" is a verb, and "a lot" modifies "worked". That by definition is the function of an adverb. Therefore, "a lot" in this context is an adverb.
ADJECTIVE: "A LOT OF"
- A lot of people go to church on sundays.
In this case, we are using "a lot of", NOT "a lot". The "of" belongs to the phrase "of people".
It seems weird, but this is supposed to say "many" people (as compared to "few"), not "an ample amount" of something called "people" (as compared to a "deficiency").
"Ample amount" and "deficiency" are values relative to an expected quantity, whereas "many" and "few" are absolute estimates that are not dependent on expectations.