Yes, did is a verb. It is the past tense of do, and it works with every subject with no spelling change.
She did the dishes last night shows did in use: last night names a finished past task, so the verb shifts from do to did.
We did our homework after dinner. Here did is the main verb: it names the action the subject carried out, so we are the ones doing the homework, yesterday or earlier today.
He did call you, but you were out. Here did works as a helper verb alongside the main verb call, showing that the call belongs to past time while call itself stays in its base form. You could also say She did finish the report before the deadline. This use of helper did is emphatic: it adds weight to the statement, insisting the past action really happened, so you would not need did in an ordinary sentence like She called you yesterday.
Did is the past-tense helper form of do, and it can also stand on its own as the main verb, naming a finished action, as in She did a cartwheel yesterday. One caution in everyday use: because doing and done behave differently from did, you should not swap them in for it. A past statement needs did, not the ongoing doing or the participle done.
They did a great job on the presentation yesterday. Here did is the main verb, naming the action the subject carried out. That is different from doing and done, which behave differently and should not be swapped in for did.
Did she finish the report? Here did is a helper verb that carries the past marking on its own, so the main verb finish stays in its base form.