>>2496694Climbing shoes are meant to be tight so you can have better control and body awareness while on the wall. If you're newer just get a half size down from your street shoes, as you progress you can do a full size down.
For cheaper shoes try the La Sportiva Tarantulace. They're about $80-90.
You'll get used to the feeling of chalk, just make sure you aren't applying too much.
When you're shorter you have less reach, but you also have less mass and an easier time keeping your body close to the wall. Focus on technique and building up body strength.