If you have lived in the same household as your spouse for the last six months of the tax year, you cannot claim to be Head of Household.  You can, however, file married filing separately if you choose to.

Am I Head of Household?

https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household

https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household

If you qualify as Head of Household, when you enter your filing status (single or married filing separately) into Personal Info, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.

We did live apart for the last 6 month of the year and my spouse took care of the two kids. In this case, based on the IRS website, my spouse can certainly file head of household. 

https://apps.irs.gov/app/vita/content/globalmedia/4491_filing_status.pdf

Now my questions are:

1. What's my filing status? Married filing separately?

2. If mine is MFS and hers is HOH, do we have to both itemize or standardize deduction, which is required by IRS for two MFS;  or my spouse can standardize deduction and I itemize deduction?

If your spouse had the children and is filing HOH, then you file Married Filing Separately.   Since she is filing HOH you are not subject to that rule about both itemizing or both using standard deduction.  But if you are itemizing----there is still no "double dipping" allowed for the amounts you enter.