the verb 'lick' with or without the preposition 'at'

"The cat is licking her (or his) paw" should be the correct sentence. Leaving it at "licking the paw" leaves the sentence too ambiguous. Is he/she licking his/her paw or someone else's paw?

I think you'd really want to say "The cat is licking his/her paw" unless the cat is licking at the paw of another animal.

If the cat is licking the paw of the other animal I'd go with the first version. The "at" isn't necessary.

I think either would be correct. If the cat has a thorn in its paw, it would quite likely be licking at it.

It's just usage, I'd say.

and note

The cat is tapping at the window. (it's tapping part of the window)
The cat is licking her paw. (she's licking the whole paw)

So does that mean :

The cat licking his/her paw means it gives a thorough lick to the entire paw?
and
The cat is licking at the paw means it licks only a part of it? [may be a wounded part of the paw or that part of a paw that somehow got smeared with milk or honey]

I wonder if honey ever tickles feline taste buds. LOL

Yes, I would say so.

But you could quote me exceptions, if you felt like it.

eg: He kicked the door.
He kicked at his opponent (but missed).