How old were Romeo and Juliet when they died? Did they marry?

In Act 1, Scene 2, Paris asks Lord Capulet for Juliet's hand in marriage. Though Capulet would obviously see the benefits of such a match, he feels Juliet is too young to marry. He tells Paris:

My child is yet a stranger in the world,

She hath not seen the change of fourteen years;

Let two more summers wither in their pride

Ere we may think her ripe to be a bride.

Juliet has not reached that fourteenth year, so this tells us she is thirteen years old. As for Romeo, there is no concrete reference in the text to his age. There are vague references to his youth, such as the one in Act 2, Scene 4, where the Nurse asks, "Gentleman, can any of you tell me where I may find the young Romeo?" These vague references have led many to put his age somewhere around fifteen or sixteen, though at least one source gave him an age as high as early twenties. Whatever the case, the lack of textual evidence means numbers can only fall in the realm of speculation.

As for the question of marriage, Romeo and Juliet were indeed married. In Act 2, Scene 5, the Nurse returns to Juliet to tell her of Romeo's intentions. After begging the Nurse for her news, Juliet finally gets her to speak:

Have you got leave to go to shrift to-day?....

Then hie you hence to Friar Laurence’s cell;

There stays a husband to make you a wife.

This scene ends shortly after these lines, and in Scene 6 the two young lovers meet in secret to be married by Friar Laurence. The Friar agrees to marry them because he thinks it might finally end the feud between their families. The fact that Romeo and Juliet are secretly married is what causes Juliet such turmoil when she learns her father has chosen Paris to be her husband. Aside from not wanting to marry someone she doesn't love, marrying a man when she is already married would be a grave sin. However, she also can't reject Paris after her father already agreed to the suit and both of her parents are pushing her into it. This dilemma is what sends her running back to the Friar to ask for help, claiming she will kill herself if he doesn't come up with a plan to get her out of marrying Paris. It is at this point that the Friar gives Juliet the potion to make her appear dead. The formation of this plan causes the final tragic moments of the play.