I believe turn, turn, turn in this song means things never remain constant. There is always change in the universe. The turning that takes place in this song represents a gradual, rather than a speedy, change from one event/time to the next event/time. In this regard, yes, turn means to change into something.
No, we do not say 'turn a season'. The gradual change I mentioned above would be summer turning into fall, winter turning into spring, etc. "Summer will turn into fall."
A time to be born, a time to die... once, again, the gradual change that takes place from birth to death.
A time to gain, a time to lose... the cycle continues as we gain something for awhile only to gradually lose it over a period of time.
Under heaven could be interpreted as God's will... but I think it means on earth. The meaning of this expression could be based upon your spiritual belief or it could be taken literally as being located below the sky.
Yes, in this song dance means to be happy and to mourn means to be sad. Yes, to dance and to mourn are common usage.
This song is one of my favorites! I hope you enjoy listening to it, also.