I am very happy to receive your email, and to know you want to get to know me more.

We usually say that we "get to know someone".

For example:
"After talking to her for a long time, I felt like I really got to know her."

I love playing sports and experiencehad a super busy time last week.

It's common to say that someone "had a time".

For example:
"We had a fun time at the beach."
"I had a hard time solving the puzzle."

So I didn't play any sports and put more time into studies last week, but usually I would like to run, play basketball and swimming.

You could also say: "I like running, playing basketball, and swimming."

I am a poor swimmer becasuse I do'm not really skilled at swim butming. I can only can jump and slap in the swimming pool, whilebut I still enjoy it.

Playing basketball usually provides a competitive feeling and game to myself,for me. I like to play against others and coprooperate with my teammates on the court.

If you ask me who do I do it with?, I would say anybody.

This should be joined to the sentence after it.

I would rather to say anybody.

(I joined this sentence to the previous one.)

Radomly is attractivI'm interested in playing with anyone.

I wasn't sure what you were trying to say in this sentence, but I hope my suggestion helps express what you were trying to say.
The spelling of the first word in the original sentence should be "randomly".

Yours

You can pick either "best regards" or "yours", but you shouldn't use both. If you use "yours", also remember to add a comma afterwards like you did with "best regards".

I am very happy to receive your email, and to know you want to know me morore about me.

"Know me more" is also used but not so often

I love playing sports and experienced a super busy time last week.

"I adore playing sports but I had an extremely busy life last week." would be better
"and" is used to relate two similar things/phrases

SoTherefore, I didn't play any sports and put more time into my studies last week, b. But usually I would like to run, play basketball and swimming.

I am a poor swimmer becasuse I do'm not really skill at swim buted and can only can jump and slap in the swimming pool, while. However, I still enjoy it.

Playing basketball usually provides me a competitive feeling and game to myself, I like to play against others and coprooperate with my teammates on the court.

cooperate = working with each other
corporate = a large company

If you ask me who wouldo I do it with?

I would rather to say anybody.

Radomlyness is attractive.

Yours

"Best regards" and "Yours" are 2 ways to end a letter so just choosing 1 is enough