With regard to having the old loan paid off, by trading the car in, your mother does not have to give permission for that - no bank in the world is going to require her permission to accept a payoff check on the loan. Whether or not she has to sign the title transfer paperwork depends on the rules of your state. Contact your state's DMV and ask them about it - thay may require that you send her a form to sign with a notary, which you could then give to the finance officer of the dealership where you are buying the new car. Some states don't require this, and your signature will suffice. Just depends on the state's regulations.
As for insuring it, most companies allow a policyholder to add any car they want to their policy, regardless of whose name it is registered in. Your mother's insurance company will have the final word, but it is common especially with young drivers whose parents carry them on their policies. Keep in mind that if you are the car's principal driver, you must also be added to the policy as well, as an insured driver, especially since you will be using and parking the car so far away from your mother's address. This may not save you as much money as you expect on insurance, and you might want to try shopping your own insurance just to see how that would turn out.
Any new lender looking at your credit file will have your application in hand and will already see that you are intending to trade in the older car, so they will not assume you will have two loans. Your application, when submitted by the dealer to potential lenders, shows the proposed deal - price for the new car, trade in value, cash down, etc - so that the lender knows precisely what they are being asked to loan you.
