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“Watch” is in the infinitive form, not the simple present form.
- I had him watch.
- I had him sing.
- I had him run.
“Watch,” “sing” and “run” are all in the infinitive form.

Grammatically, “had” in your example is being used like “let” or “made.” You could change “had” to “let” or “made” and the sentence would still be grammatically correct, although the meaning would change:
- I had a cute cat watch me play.
- I MADE a cute cat watch me play.
- I LET a cute cat watch me play.

> 2, according to my knowledge in English, shouldn’t “watch” be “to watch”，or “watching?”

When “had,” “made” and “let” are used in this way, “to” isn’t added in front of “watch,” even though “watch” is in the infinitive form. But you could change “watch” to “watching” without changing the meaning of the sentence very much:
- I had a cute cat watch me play.
- I had a cute cat WATCHING me play.

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When saying 'I had' this way, you are setting yourself up as the topic of the event that happened to you which would otherwise be the second subject (the cat.)

*A cute cat* watched me play drums.
*I* had + a cute cat watch *me* play drums.

As to why it's in the present present tense, I don't know. It's an unwritten rule that 'subject had a noun + present verb + object pronoun' is the way it is.

'To watch' doesn't fit because it isn't apart of I had. 'I had to watch the news' means you needed to watch the news like a chore or you couldn't resist watching the news.

'I had a cat watching me play piano' is correct and is similar to the original sentence. To make it easier it's kind of like this: I had ± a cat (who was) watching me play piano.