The key to answering your questions is determining which of you is the custodial parent since it is only that parent who can file as Head of Household if otherwise qualified. The custodial parent is the one with whom the child spent the most nights. In your case it appears that would be the mother for 2021. If one child lived with you in 2022 and the other with mom and you had separate households then each of you could file as Head of Household in 2022.
Note that the amount of financial support provided to the children is irrelevant.
Property tax and mortgage interest deduction is deducted by those who have the mortgage and own the property. In your case, if that is both of you, particularly if you are in a community property state, you would split the deductions.
One thing to keep in mind, and it's important.
If you are still married on December 31 of the tax year, and you file with a filing status of "Married Filing Separate", then neither of you qualify to take "any" education credits or deductions, regardless of who claims the dependents if they qualify to claim them.