Noun Use

Hello Sir

Thankyou in advance

Please verify the following

1. My confusion
* if a verb (non finite) has an object then we either use gerund or infinitive, that's why it's wrong/incomplete to say

Eg. I didn't see him playing.
It's better to say- I didn't see him play.(since there is no object)

Or Complete sentence would be:
I didn't see him playing football

Is that right?

2. In above sentences, word 'play'- is it a verb or a noun?? Is gerund a noun?

3. For words which are both noun/verb do they also have deverbal noun form? If yes when would that be required if we can simply use the word(noun form) itself when noun is called for?

Thankyou sir