So I want to know whether the "doing sports" is a participle or a gerund. I have looked into both and both seem possible to me although I think a gerund would make more sense with the addition of "with" for example but maybe you can tell me why this isnt necessary. I normally can dintiniguish the 2 quite well the constructions with "spend" and "waste", however, I can not give an answer as to which one it is. Another one not containing either of the two : "While I could get by not knowing what it is I want to know because it helps me understand the language better." So is this a participle? If I added "with" in both these 2 sentences would they become gerunds? ( "with doing sports" "with not knowing what it is")
Some writers on grammar don't accept that there is such a distinction, and prefer to say that the "-ing" participle can be used in different ways.
But in traditional terms, I think it has to be a participle, because that slot in the sentence can only be filled by adjective-like phrases:
I spend a lot of time asleep.
I spend a lot of time drunk.
and not by noun-like phrases:
*I spend a lot of time history.
*I spend a lot of time recreation.
(But on the other hand "I spend a lot of time on history" is fine).
Side note, not directly relevant to your question: in British English "I spend a lot of time doing sport" would be more natural. You could say "sports", but that would be emphasising that you did several different sports. I believe that in American English, "sports" is the normal form for "sport in general".