Head of household:
To file as head of household, the taxpayer must pay more than half the cost of maintaining their home, and they must provide care for a qualifying dependent (usually their child) in that home. Earning the most money doesn't necessarily mean that person pays more than half of the cost of maintaining the home. The person who earns more might might be supporting other children, or their parents, or saving their money. But it probably is true most of the time that the person earning more money pays more of the cost of maintaining the common household.
Mortgage and property taxes:
To deduct the mortgage and property taxes, you must be an owner of the home, you must be legally obligated to pay the expense (listed as an owner and/or listed as a borrower) and you must actually pay what you want to deduct. If you pay 30% of the taxes and interest, you would normally deduct that amount of the expenses on your tax return.
The IRS will usually not dispute whatever arrangement you decide. For example, maybe you decide that your partner pays 100% of the mortgage and property taxes, but you pay 100% of the utilities, insurance, and food for the home.
The person who claims the mortgage interest and property taxes is not required to also be the person who claims head of household. They aren't connected, except that the person who claims HOH status must pay more than half the total household expenses. If partner 1 claims all the mortgage interest and property taxes, but partner 2 wants to file as HOH, partner 2 will need to document that the other expenses they pay (food, insurance, repairs, utilities) are more than partner 1 paid.
Disputes:
If you can't decide which partner should claim the child as a dependent, the IRS tiebreaker is whichever partner has the higher income. If you can't decide how to split the mortgage interest and property taxes, each person can only claim what they can prove they paid. If the expense was paid from partner 1's account, and not a joint account, partner 2 may have a hard time proving they paid a portion of the expenses.
Ultimately, if you can't agree how to file your tax returns, you have a significant relationship problem that's more important than your tax returns.
