However, as I've written before, Russian winters aren't particularly conducive to long walks, so I have to stick with indoor activities out of necessity.

That means regularly going to the gym and to the swimming pool regularly.

There is nothing wrong with where you put the word. It sounds more natural to me at the end. It also follows a logical progression: the main action comes first, followed by the adverb of frequency.

What I really like about these [sports] is that afterwards I am pretty much worn out, which, [paradoxically], is very pleasant.

"Going to the gym" isn't exactly a sport. Maybe there is a different category that both activities would fit in. Perhaps "types of exercise," or "physical activities," or maybe even "workouts," if you wanted to focus on what you do in each location. I agree with @thomas256 about "paradoxically."

If you're worn out, you basically have no more energy left for stress or anxiety, and you can finally haveenjoy some well-deserved rest.

"Basically" and "more" are redundant.
"Well-deserved rest" is I what would suggest using here. The phrase is so common, it sounds incomplete without all three words. It also tends to get paired with the verb to "enjoy."

However, as I've writtenmentioned before, Russian winters aren't particularly conducive to long walks, so I have to stick with indoor activities of necessity.

"Mentioned" is a more general term. You could add "in a previous post."

Both "of necessity" and "out of necessity" work here, but there are some minor differences. One difference is that "out of necessity" is used in spoken language and the kind of informal writing we post here, whereas "of necessity" is more formal. Another difference is that "out of" points to the reason why someone has to do something, whereas "of" points to the circumstances that make something inevitable.

As a side note: "out of" can be used for a range of different reasons, such as necessity, fairness, friendship, curiosity, ignorance, negligence, a sense of duty, a sense of responsibility, consideration for.
and a abroad range of emotions, such as anger, jealousy, despair, greed, spite, compassion, kindness, generosity.

Maybe you could recommend me some?

"Me" isn't needed.
"Suggest" would also work here and be a little less formal.

Alternatively, you could end with a direct question:
"Do you have any recommendations?"
"Do you have any favorites?"
"Do you have any suggestions?"
"Do you have any to suggest?"

That means regularly going to the gym and to the swimming pool.

Second "to" can optionally be left out here.

What I really like about these sports is that afterwards I am pretty much worn out, which, paradoxicactually, is very pleasant.

"Paradoxically" doesn't quite feel right here. I would use "paradoxically" to describe a nonsensical situation which seems to logically contradict itself in some way. Like if being happy made you really sad. But being worn out and feeling pleasant don't seem like direct opposites enough to warrant a "paradoxically".

However, as I've written before, Russian winters aren't particularly conducive to long walks, so I have to stick with indoor activities out of necessity.