"I celebrate my birthday today" is correct, but it sounds like a formal declaration. Imagine that you are standing on a high balcony, with thousands of people eagerly listening to you. You spread your arms and shout the sentence to the heavens. That's what I'd call "an appropriate situation". :) We typically say "I am celebrating my birthday today" because your birthday is not an everyday event. That sounds normal, and can be used in almost any situation.

Yes, both are correct, but I see the first one more often ^^

Tense! They have different meanings. Do you remember your tenses? The simple present tense is for a regular occurrence. "I celebrate my birthday today" means "My birthday falls on this day every year". It has nothing to do with whether you are having a party today. The present continuous tense may be used for events in the near future. For example, "I am taking an exam this afternoon." "I am celebrating my birthday today" means "I shall mark my birthday with some kind of party or celebration today." It does not necessarily mean that your birthday falls on today's date. Thus "We are celebrating the Queen's birthday on 11 June next year." In fact, the Queen's birthday is on 21 April. The official celebrating of her birthday takes place every year in June, for better weather, in a ceremony known as Trooping the Colour. So you see, everything is more interesting than they seem. Tenses are also extremely important.

When someone says "I am celebrating my birthday today" means a continuous state which is the state of celebration and we can see that because "celebrating" was used. Now, when someone says "I celebrate my birthday today" means that that person stated just a fact. The both are correct and in my point of view I do not see one to be more formal than another so I encourage you to say what you want to mean, state of celebration or just a fact ?

The first sentence is more correct than the second, because it's a temporary event. :)