Can I ask to have additional money taken out of my distribution so I do not have to owe as much taxes when I file taxes? 

Sure.  Ask the plan.   Or send in an estimated payment now.

Here are the blank estimate forms and instructions for federal.  You need to go to your state's website to get theirs.

https://www.irs.gov/pub/irs-pdf/f1040es.pdf

Or you can pay directly on the IRS website https://www.irs.gov/payments

Be sure to pick the right kind of payment and year.....2019 Estimate

Remember state taxes as well.

How can I get a copy of my 2018 taxs return

Guess you are lucky we check old threads sometimes, since you posted a completely unrelated question to an old thread  about a retirement account.

PRINT RETURN   You need to print a copy of your return?  Sign in to your account using exactly the same account and user ID that you used to prepare your return.
On the bottom of the screen find Your tax returns & documents and click on Show.  Click on the Year and Click on Download/print return (PDF)

Have a situation where, I was laid off - my 401k was directly distributed to me instead of being rolled over to a qualified plan. Taxes were withheld - plan on rolling over the check I received - how do I get a credit for my taxes that I paid so I can roll then over as well?

The 2019 tax withholding will be credited on your 2019 tax return and any of the tax withholding in excess of your 2019 tax liability will be refunded with you file your 2019 tax return.  In the meantime you'll need to substitute other funds to complete the rollover of the entire distribution since the rollover must be completed no later than 60 days following the date of receipt of the distribution from the 401(k), long before you'll receive any refund after filing your tax return.

You had the option to tell the 401(k) plan to do a direct rollover to another qualified retirement account, say, an IRA, and avoid having any amount withheld for taxes.  Since you did not do that and instead received a distribution paid to you, the plan was required to withhold a minimum of 20% for taxes, putting you in the situation where you need to substitute other funds to complete the rollover of the entire distribution by the deadline.

If you don't replace the 20% taxes with your own money when you deposit into the new account....then the withholding will become a taxable distribution itself.

Is the 10% penalty automatically calculated when filing using TurboTax?

I took a total distribution of my 401k and they withheld 22% including for federal and state. 

Does TurboTax calculate and add for the penalty in the final figures of our refund/debt?

Yes.  

You didn't actually pay the tax or 10% penalty (you pay a 10% early withdrawal penalty if you are under 59 ½). You had taxes withheld like from your paycheck. You still have to enter the whole amount (before taxes were withheld) with your other income to figure out the total tax (and it may put you into a higher tax bracket) and then the withholding is subtracted from the total tax to figure your refund or tax due. 

It has to break out and list the 10% early withdrawal penalty separately on your return. It's on 1040 Schedule 2 line 6. Then you will get credit for the withholding on 1040 line 17.

And it's very very common to owe more because the withdrawals increase your income and put you into a higher tax bracket. So when you add all your income for year you didn't have enough withholding taken out of the 1099Rs to cover the total tax. You owe more than they sent in for taxes.

And by increasing your income it would have changed other items on your return like make more of your SS taxable or changed the limits on your Medical deductions or employee job expenses and changed any credits you qualified for.

Ps…..you can lose up to 50% of your withdrawal to federal and state taxes and penalties.