Here are the correct forms of the verbs in the sentences:
How often he plays (to play) guitar?
I am not writing (not to write) a letter now.
Are you going (to go) to the south next summer?
Where are you going (to go) next summer?
Where are they going (to go) on Sunday?
They are playing (to play) in the yard now.
What did your brother do (to do) yesterday?
Where will you go (to go) next morning? (Note: "next morning" suggests future, so "will go" is appropriate)
He sleeps (to sleep) every night.
They are not playing (not to play) in the street now.
Let me know if you'd like explanations for any of these!