Is it appropriate to say that my birthday comes (or arrives) in May, Or Christmas comes in December? Which other alternative can I use instead of comes or arrives?
Could I also say, "my birthday 'falls' in the summer season"?
Using comes in that way will be understood, but it is far from idiomatic. The most usual way is just to use an appropriate for of to be:
My birthday is in May.
(It's not, but what the hell)
Sometimes, like if it was recent and you're highlighting the fact that it will be a while until it comes again, you would use the past tense.
Easter was in March.
Falls can be more idiomatic, but is is still safer. Arrives should not be used in such situations - it will be understood, but seem very strange.