Thinking about getting a new car?

You need to realize that lenders require no less than six months of full-time employment history, and a minimum verifiable income of at least $1700-$2000 per month, in addition to sufficient credit/credit history. Not meeting income requirements is going to get you denied automatically.

In addition, you'll need to obtain full coverage insurance, which will be just as much as, if not more than, your payment. You'll also have maintenance, upkeep, and gas to pay for.

Sorry it's not what you wanted to hear.

There is absolutely no chance that you would qualify for any kind of car loan or financing. Your income is at not even half of the minimum amount to get any kind of financing.

You're way over your head. Save all of your income for a year, and then you'd have $8,000 cash with which to buy a used car and insure it.

Your only option is to buy one in cash. Your income is too low for financing.

Traveling for college, what are the specifics on that. If you are living on campus, then just get rid of the car and get rides back home for holidays or take bus. Having car on campus is a waste of time and money.
If you are commuting, then I think this car is OK.
Do not go into debt, and you would not qualify for a loan.
So sell the car and save up for another one or fix this one.
Sure you will have to put some money in it. Also try to shop around more on the repairs. Maybe find a smaller shop that will work with you on the price. $350 for 1 wheel bearing seems high, not sure on that. In my area, there would be NO WAY for you to use any type of national chain. Their prices have gone way up over the last few years. A national tire store in my area charges the same hourly rate as the local Audi dealer.
Find a body shop to clean some things up. They can reattach the fender properly with new clips. I am guessing that would be $75. And maybe pull a few dents out and buff out some areas, it might look better.
As for not starting. That might be an easy fix, do some internet searches.
Change the air filter and spark plugs, you can do that yourself. Change the fuel filter if is over 3 years old.
One thing you might have to think about is the timing belt. That may be past due for a change. That might be around $900, but has to be done at least every 100,000 miles.

You don't don't any options. The PT Cruiser was a junk car even when new, so it's not unusual that you are having so many problems. But unless you have cash to buy a new car, you are not in a position to get a loan for a better car. But when you do buy another car, do better research so that you will do better than a crappy PT Cruiser.