They may deduct what you owe

You are getting taxed @ 10% on the first $9525 that you earned and 12% on the next $9526-$38700. However, considering that the standard deduction for a single person (no children) is $12200 for 2019, your adjusted income will only be $16600 - $12200=$4400. You will owe $440 in income taxes plus $260 in penalty for a total of $700. Check your old pay stubs to see how much money your employer has already withheld for Federal income taxes. 

By the way because your 401(k) money will count as income, you won't qualify for EIC (Earned Income credit, "free" money from the government when you file your tax return) because it pushes your income above the limit for a single person without children and you will lose that money on top of having to pay penalty for early withdrawal of your 401(k). 

How old are you? If you are under 24 and not counted as an independent student, your higher income will also impact you when you apply for FAFSA. Your Pell Grant will likely get reduced. 

You add it to the rest of your income and pay any income tax as well as the penalty. Because you didn't make very much this year, it shouldn't be too much. You also might be eligible for an education credit that would cover part or all of it. They should withhold at least the 10% penalty, but it is more common to have 20% withheld. When you file your taxes any overpayment will be refunded.

Just open an IRA account and transfer the 401K there.  Then put it in a S&P 500 index fund and let it grow, some day you will be 40 years old and be happy you did that.

2600 is nothing.  Don't listen to people who say to leave it there.  They don't understand how money grows.  That amount might double in the next 8-15 years.  So in 10 years, maybe - you'll have 5200 in there.  And if a recession hits before then, then you might have 3000 in there 10 years from now.

It's not going to affect your taxes much, at least not federally.  Depends on your state.  I'd take it out and pay off a monthly obligation, freeing up real money.

Yes, you'll pay income tax on the withdrawal plus the $260 penalty.

Ignore their letter.  They just don't want to be bothered to keep sending you statements.  Don't worry about that - it's their responsibility to do so, unless and until you get a 401k elsewhere and transfer/roll over that account to your new one.  Yes, it will affect your taxes.  Leave it there.  It will accrue and be worth considerably more money in years to come.

Yes, in addition to the penalty, you also pay regular income tax on the amount withdrawn.

Yes, it’ll affect your taxes. It’s considered income so you’ll have to pay tax on it. 

Your best option is to roll it to a Roth IRA.

You will pay a small amount of taxes but if you have a standard rate of return (double every 7.5 years) you will have more than $80000 in 40 years = tax free.