It goes away after a while. You get dizzy because your body isn't used to having you blow so much air like that. Maybe lay down for a bit. When it goes away, start playing again. Your body will grow accustomed to blowing in the flute and you won't get dizzy as often.
All people have their arms get tired.
Never play at an angle because your tone will be sacrificed. The flute is meant to be played with the head straight and the flute straight. A lot of flute players play at an angle, but that doesn't mean you shouldn't. If you play with a straight head and flute, people will notice because you stand out and look much better. People like your band teacher.