Is it a participle or a gerund?

Question

I made two sentences with a word I intended to use as a gerund. But I noticed that it seems like a participle, too. Can the word "having" I wrote below have the both roles?

1. There are some points to think about having a pet.
2. We have some points to think about having a pet.

(As a gerund: About having a pet, there are some points to think.)

(As a participle: Having a pet, we have some points to think about.)

Or can I say like this?

3. There are some points to think about of having a pet.

I would like to put the sentence after this sentence: What does having pet mean? (about responsibilities when you have a pet)

Answer 1

Sentences 1 and 2 would be ungrammatical if "having" were a gerund, because it would be the object of "about". In that case, "points" would be the object of "to think",¹ but that would not make sense. (You can't "think points", at least not in the usual sense of that verb.)

Sentence 3 could work, but "of" would not be the usual preposition there. (Some people might prefer "regarding" or "concerning". You could also say "when owning a pet".)

Perhaps it would be best simply to rephrase the sentence. For example:

If you have a pet, then there are some points to think about.

However, if you want to put it after a sentence that already mentions "having a pet", then there is really no need to repeat that information:

What does having [a] pet mean? There are some points to think about.

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Note 1: I'm being a bit loose with terminology, but I don't want to get into too much detail.