diving into modals

no verb agreement	He swims but He can swim. *He cans swim.
never followed by to or a verb with -ing or -ed/-en	I want to swim, I am swimming, I have swum but I can swim, I can to swim, I can swimming, *I can swum.
can not follow to have -ing or -ed/-en ending	I want to can swim. To can swim is wonderful. After canning swim, he went home. He is canning swim. *He has canned swim.
can not be used with do to form questions or negatives	He does not swim. Does he swim? but He does not can swim. Does he can swim?
the past tense form usually signals present or future activity rather than past	He might swim this afternoon. He might swim yesterday. He should swim this afternoon. He should swim yesterday.
are always the first auxiliary before the verb. Can not be combined with another modal.	He may have been swimming. He have may been swimming. He may can swim. *He might could do it.

Some suggest that the following are modals. Do they fit all of these criteria?
better, best, dare, need, ought to, have to (hafta), going to (gonna),