Hmm, I don't think there's native functionality to be programmatically alerted when a comment is created I'm afraid; I skimmed the webhook docs to see if it's possible and it's never explicitly mentioned as not being possible, so maybe it's worth testing? 

I'm wondering if it'd be possible to have an account be subscribed to every record so that every comment sends you an email, and you could use some other service to parse the link in the email to get the record ID.  You could then use that record ID with the List Comments endpoint to grab the latest date / comment count and then update it into that field (or just trigger the webhook directly instead, really)

Would there be a possibility, using a formula field or a script, to update a field that would count the number of comments or retrieve the date of the latest comment to avoid searching through emails, etc.?

I don't think formula fields can access comments at this time I'm afraid

Scripting's doable, you'd use the List Comments endpoint, but the problem is we don't have an automation trigger for when a comment's made, you know what I mean?  You could get the script to run once an hour or something to grab the comments for all of your records maybe?  That'd be 720 automation runs, not too many actually