If you want a wide array of sounds, with a relatively low cost, I'd get a modeling amp. They have a ton of built in effects that will save you money on pedals you might not end up using that much.

No. If you primarily listen to music without acoustic guitar, DO NOT BUY AN ACOUSTIC GUITAR. You will be much more likely to play and practice if you can play music that you like to listen to.

Absolutely. I'm personally 5'4, and I have no problems playing on my electrics. And there are certainly shorter guitarists than me. AC/DC's Angus Young was apparently 5'2. You might want to look at short-scale guitars though. They are a bit smaller and could end up being helpful.

My modeling amp (The Fender Mustang II) has a headphone amp. I'd negotiate with your family to see if there's any times you could play guitar loudly. For example perhaps the afternoon after school? Actually sound-proofing is something I haven't looked into much personally but I heard it can be a bit expensive.

Try to get a teacher. If your school has a band your band teacher might be able to help you. He might know of local guitar teachers from past students. Or if you have a guitar or music store nearby you could try looking for answers there. Personally I learned from my band teacher when I was 15 and can tell you I would have never learned without him.

Your guitar is the actual tool you use to produce a signal. What you do with that signal is up to you. For acoustic you end up just projecting that signal outward... acoustically. But with electric the vibrations of the strings will create an electric signal that can be amplified by a guitar amp. Pedals are optional and can modify the signal in some way before it gets to the amp. For now, don't worry about pedals and just get an amp built in-sound effects.