No., Unless it is a special type of loan, which would have been issued on a debit card, and you receive bills and pay directly to the company every month. (access advantage, formerly reserve loan)
This is not all that common, if your payments come directly out of your paycheck, that is the only option for paying back the loan. The agreement was between your payroll and you, not the administrator. If you leave the company, you must pay the loan back in full within 2 fiscal quarters or it will be deemed a distribution and subject to tax and penalty.

Since the economy has been strapped, 401k rules have been changed. The plan administrator can explain any changes that may ahve affected to plan. Must employers cancel a 401K plan when an employee resigns, sends them a check and deducts loans, taxes, fees, etc. This has been found to be illegal unless the 401k is set up to invest in the company only, which 99% of them are not. If you resign, the company stops all contributions and turns your account over to the administrator or investment company. Depending on how the plan is set up, the investment company may be obligated to close the account. You should have a converstion with the investment manager before maaking any decisions to see if you can roll it and the loan into an individual IRA with the investment firm.

Maybe but probably not.

Your first question is if you can keep your plan at the investment company. It depends on your balance. If your balance is greater than $1000 then, yes, you can keep it there. If it is less, the record keeper can force you out of the plan, meaning you would have to take your funds elsewhere.

If you are forced out you could possibly rollover to the same investment company but it is highly unlikely that a rollover to any other source will accommodate your outstanding loan.

If you do keep the funds in the current plan with that investment company your second question of whether they will let you make loan payments is "it depends". Some will, some won't. It is the most common practice that they won't allow this as they only want payroll deductions from the employer, and since you no longer get a paycheck that is not possible. However, there are investment companies that make exceptions and will allow checks directly from a participant in this case so the only way to know for sure is to ask them.