I'll answer them one by one:
I've always wanted to play an instrument, and loved music, though I've just never had the time, money or motivation to do so. I'm 17 and will be starting my final year of high school in about 5 weeks - but until then, I've got some time to spare: you can learn violin at any age, i started when i was 35 and found it easy (despite being a dumbo at the piano in my teens).
And after seeing Lindsey Stirling play (she's epic!), I searched ebay and realized I could get a cheap violin for like $60. - NO, NOT ON EBAY. go for one of the used ones from shar music. http://www.sharmusic.com/2010-Shar-Email-Offers/Outlet-Sale/Previously-Owned---Moderate-Wear---Franz-Hoffmann-Amadeus-Violin---Instrument-Only---1-2-size.axd NEVER EVER BUY A VIOLIN ONLINE FROM RANDOM SELLERS ON EBAY. the link i have put in above is for a half sized violin, but if you talk to the shar people, they will find you a violin to fit your budget and your requirements.
I wouldn't be able to afford a proper teacher though, and I'm worried self-teaching will be too difficult or frustrating? - self teaching, even if from very good teachers on the internet, will only make you learn bad habits that are hard, if not impossible, to unlearn later. however, internet teachers make very good supplements to a real teacher.
And I don't exactly have anywhere isolated to play - 7 people live in my house and I don't want to be annoying them with my horrid attempt at learning a loud, high-pitched instrument LOL. I learned to juggle with tennis balls though, so I'm fairly dexterous, and I love the sounds violins make (when played well) - i wouldnt mind annoying them! plus you can always fix a mute to the violin bridge and mute the sound. mutes are very very very cheap.
So:
1. Is it difficult? How fast can I begin to sound decent? - i could play twinkle twinkle little star in 2 months. and it only gets easier after that.
2. Is getting a teacher necessary? - you have to. there is no compromise on this.
3. Is it possible to buy an electric violin with headphones, so nobody can hear it? (is that more expensive than an ordinary violin?) - i think it will be more expensive. and it is always better to start with an acoustic and then move on to electric after getting thorough with the acoustic.
4. Any ideas on isolated places I can practice? - fix a mute on your violin.