"I like exercising on weekends."
Yes, it nominalises (makes a noun) the verb 運動する (to exercise) so that you can say that you 好き (like) it.
If I remember correctly, you dont use a は there because you are talking about general time (every weekend/most weekends) instead of a specific weekend.
Someone correct me if I'm wrong please.
I suspect that the reason why Duo doesn't allow は here is because it assumes 私 as a topic in this sentence, forming a classic example of "wagabun", 私はXが好きです. Therefore, the full sentence with は would be 私は週末は運動が好きです and would have two topics, which is not outright forbidden by any grammatical rule but is usually used only in special context. But Duo deems it unacceptable.
Imho, Duo should accept solutions with は since it accepts other borderline translations anyways, and there is no context, so we don't know if the whole conversation doesn't create a contrastive pattern, for example. But that's just my opinion.
Duo is really pedantic about translations, I guess を isn't accepted because then you are not really saying "exercising", you're literally saying "doing exercise".
As for the は after 週末 that's definitely acceptable and should be reported.
No. The の here isn't possessive, it nominalizes 運動する. Without it the sentence would be grammatically incorrect.
It was allowed in classical Japanese that the 連体形 of a verb can act as a noun directly. Examples which are still used in modern Japanese are するがよい, するにちがいない. These are memorized as a fixed phrases and do not require こと or の to normalize the verb.
運動 is already a noun for exercise. So surely you don't need to add する and then nominalise it with の?
I'm pretty sure Duo has done this in the past for する verbs.
運動が好き has a different nuance as 運動するのが好き. The latter one focus on "doing" the action while the former one does not. For example, it may be that you like to talk about sports or watch others to do sports and still say 運動が好きです.
When should you put a particle (は or に) after a time expression? (e.g. yesterday, on the weekend, last year etc...)