Is it possible to not be in control of a character that I have in play? If so, how is this handled?

This is quite possible as a lot of card effects create this scenario.

Once you play a character and loose control of the character your opponent who controls it can do more or less what they want with it, including but not limited to attacking, defending, using for claim, triggering abilities and paying costs.

If the character is unique you cannot play or put into play another copy of that character.

If the mechanic that took control of the character is removed or overwritten then control goes back to the original player (or in the case of lasting effects i believe it would reassert the last effect, although on this point im not 100%).

For example, the lannister attachment 'enslaved' takes control of a character of printed cost 2 or lower, if this attachment was subsequently removed or blanked then the character would be returned to the control of the person who played it (other effects not withstanding.)

"You" or "your," when used to refer to cards in play, always refers to cards you control. You must control a card to do pretty much anything with it. By default, you control all cards that you own/play/put into play (including attachments you put on other opponents characters). Control only changes from this default when a card effect specifically says otherwise. Examples from the core set include Seductive Promise and the various Honor Guards.