I love the concept and glad that this is a demo because it does need a bit of polish. As it stands right now, I wouldn't spend money on something like this for anyone I know (unless they specifically asked for it) and I'd probably be disappointed/bored if I get it as a present as it stands now.
For the actual game and game play. You've already been told that the controls are kind of clumsy. I feel that it's more, you're just too big. Especially when some of those fireballs come at you, you're barely the right size to get in between them, it's way to easy to hit one, even right off the bat. Either they need to be a little smaller, or you do.
You also look pretty silly when you fly up, the neck gets all stretched out weird. Doesn't look good.
Things that could improve the game:
Hit points. It'd be nice to survive if you get hit. Three is pretty standard. You could add items that replenish health as pick ups.
Levels or Check Points. After going so far getting a "Congratulations! You've gone XXX! Press F to continue" or adding in Levels and having to collect X amount or go X distance each level. As this is a birthday present, it'd be nice if you could get a 'Congratulations!' now and then instead of the end always being 'Game Over'. I wouldn't want something as a gift that would make me feel like I'm always going to lose.
Achievements are always something good to have in a game, even if it's in game only. "It's my party" - Die X amount of times. "Spoiled" - Collect over 100 presents. "Sweet Tooth" - Collect over 200 cupcakes, ect.
Submitting more then one picture of said friend/family member for the game. Having the only one face made certain parts weird. It can work for those dancing e-cards having one face but they're usually short. The 'SO HAPPY' face didn't really fit with the dizzy/knocked out 'Game Over'. Either that or have an over-layer. Such as dizzy eyes and maybe a sad/dazed mouth to fit the situation.
A return to main menu option would be nice too.