An error my ex husband made on his taxes have caused my taxes to be rejected. Once he amends his taxes then can I resubmit my taxes as is through TurboTax?

Why did yours reject?  Did he enter a ssn on his return that he wasn’t suppose to?  If he claimed a ssn then you will have to mail your return this year.  Once a ssn has been used you can’t efile it again even if he amends his.

He listed her as qualified under Head of Household, but did not claim her as a dependent. I haven't seen his whole return so I don't know if he had to enter her ssn too. I believe this nullified my Head of Household status when I listed her.

Only the parent that the child physically lived with more than half the year can claim the EIC, Child Care Credit and HOH filing status.  If that us you, then you will have to mail file this year.  The IRS will not allow any other tax return to be e-filed with the same SSN, reguardless of an amended return or not (amended returns also take 4 months or more to process).

She did live with me and qualifies under my HOH. If it takes four months to amend his taxes what does that mean to my taxes if I mail them in before the April 15th deadline? Should I file an extension now?

If you mail before April 17 you do not need an extension.

You can file now.  The IRS will not match returns for a year of so to allow for amended returns to be processed.

Even if amends, it won't clear the e-file block on your return.  You should print and mail your return with the correct dependent(s).  Don't forget to mail your state return as well.