K-1 is not going to be received by April 15th.

1- File my tax return and amend this when I receive the K-1. This issue with this is I believe I will owe more taxes but I don't know the exact amount that I need to pay by April 15th. Will I be penalized when I amend my taxes once I receive the K-1?. Ask the partnership can provide an estimate of the income you'll need to report. then include it in Turbotax. if they can't or won't use your best guess to avoid penalties for underestimating your balance due. Full payment is due 4/15. There would be no penalties if you file an amended return after getting the actually k-1 and amending if nothing is owed. I recommend option 2. If there is an overpayment showing on the return filed the iRS will refund this so you could be short when the actual k-1 comes through.

with an extension, nothing is refunded until the actual return is filed.   

2- File an extension. Again I don't know the amount I will owe on April 15th because I don't have the K-1.  Will I be penalized once I receive the K-1 and complete my taxes? Same as above file an extension but you need to include an estimate of the income for the partnership in calculating if there is a balance due that must be paid by 4/15. Underestimate it and the IRS will charge you late payment penalties and interest. Then when you get the k-1 you change the numbers and e-file 

3- Create an estimated K-1 without knowing the actual number of the K-1 to try to cover the amount I will likely owe. How do I show this as an estimated K-1 in Tur/that's how you get the estimated income into Turbotax in the correct section. 

Turbotax does not allow an estimate deignation for a box when e-filing so you'll have to enter it as an actual amount or file by mail.