Hi Romero2022,
Good questions. Based upon your information posted, it would appear that you each would qualify for HOH. It is clear you not married so that is non-issue. If the kids meet the child dependency or even qualifying relative status, then HOH would be each of your filing status. In regards to the mortgage that is a bit tricky because without know the amounts you want to split, the standard deduction may be more tax beneficial than the itemizing. As of 2017, new tax laws changed and more tax filers were now claiming the standard deduction based upon their filing statuses. I will send you a link and hopefully this can help assess 2022 tax filing. See below.
https://www.irs.gov/newsroom/why-its-important-that-taxpayers-know-and-understand-their-correct-fili...
also included will be the standard deduction tables as well. That will provide good resource to what tax benefits for you and your fiancee. Congrats on the home purchase.
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