First, nuke isn't your rate. A rate would be MM (Machinist Mate), EM (Electrician Mate), ET (Electronic Technician). Nuke is simply the general field that you will work in.
1.) During basics am i allowed any contact with the outside world? ex: cell phone.
You can write letters (snail mail), not sure if they allow you access to email (I went to boot camp in 1991), no cell phone.
2.) I've heard nukes make a lot of money compared to other rates is this true and if so how much money do they make?
How much money you make is based off of your rank. Talk to your recruiter about how the rank structure works. As a nuke on board a ship/sub, you'll receive a little extra money each money (when I retired in 2011, the supervisor level nukes got $450 extra a month, before taxes)
3.) What are the living arrangements for all the schools, and are the schools 7 days a week or are weekends free?
Single, you live in the barracks. Married, you live base housing or our in town. Weekends are "free" in that there is no class. Homework and studying? That's a different matter.
4.) Are cars allowed at any of the schools?
If I recall correctly, no cars until Power School.
5.) Are you allowed to take leave when in any of the schools?
During the duration of the actual school, no unless there is an extreme family emergency. What the Navy defines as extreme for this scenario is probably a little more severe than most civilians will expect.
6.) When if at all are personal electronics(Laptops, Cell Phones, video games) allowed.
You can have that stuff in "A" school and usually aboard ship. However, cell phone use while the ship is underway (carriers) was strictly controlled when I was still in. Might have changed since, but not I'm betting on it.
7.) What is carrier life like when at sea( living arrangements, meals, time off ship at various ports, free time activities)
you sleep in a 6 1/2 foot, by 2 1/2 foot, by 2 1/2 foot box, meals are served 4 times a day, you'll probably get 3 day port visits on deployment, with maybe a day and a half for going out in town, until you are fully qualified all of your assigned duties, free time is not a luxury you are afforded.

8.) If my stationed carrier is at its home port and not deployed can i live off base and receive a rent allowance?
Depends on your rank. Ask the recruiter.
9.) What do i have to do to be able to pick where I am stationed or the ship I am assigned to?
You don't get to pick. You get to tell the Navy what you want as far as specific ship and/or port. Then, the needs of the Navy will determine where you end up (get used to seeing and hearing that phrase, "Needs of the Navy")
10.) What are the singing bonuses for navy nukes and when do we receive them?
Definitely a question for your recruiter.
11.) Is there a lot of opportunity to travel while in?
LMAO (sorry), the Navy will travel you to lots of places. If you mean traveling while not underway, depends on your ship's in-port work schedule, how much leave you have built up and how good of a chain-of-command you have.
12.) What are practical officer routes if i am going enlisted nuke?
1st question; Why do you want to be an officer? If the answer is because of better pay, you're already doing it wrong. Google "Navy officer programs"
13.) Is life outside of work while ship is not deployed relatively normal?
NOTHING is ever normal once you join, unless you consider being at work before most of the rest of the civilian population is out of bed, staying at work overnight once every 3 - 4 days (standing duty) and then working a full workday the next day, being told what to wear everyday, when to shave and get a haircut, etc. What you'll find happens is that you will come to define "normal" differently.

14.) Do you have a lot of spending money?
Depends on so many things, specifically how much debt you already have, how much debt you get yourself into and how disciplined you are when it comes to budgeting and saving.
Yes, there are a lot of things here you should ask your recruiter about, that's part of their job. Hopefully these few answers are somewhat helpful. They by no means are complete, so you still need to talk to your recruiter.
Best of luck!