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Yes and thanks for writing back to me. Here is my opinions and recommendations.

1. Since your daughter and son-in-law were married after the first of this year, it would be best for them to File Married Filing jointly to take advantage of all tax credits including child tax credit, child care dependent credits etc., full itemize deduction amounts, or increased standard deduction amount, if unable to itemize. I realize this divorce may not be amicable but if it is, your daughter and son-in-law could split the refund or share the burden of taxes due if this is the case. Also this decision may be best for the goodwill of the children and promote a degree of civility, which is vital for the well-being of your grandchildren.
2. Neither her husband or herself can claim head of household  because one of the requirements for Head of Household is that they need to be considered "unmarried" at the end of the year. One of the requirements to be considered unmarried is that a spouse didn't live in your home during the last 6 months of the tax year. Since you mentioned they separated in October, then neither would qualify for Head of Household.
3. Now, the only other option is to file Married Filing Separately. The standard deduction for this filing method is $14,600.  Now since she is entitled to 1/2 of the mortgage and 1/2 of property taxes and if she has other itemized deductions that exceed $14,600, she may itemize.  Legal fees are not deductible for divorce settlements.
4.  If you file Married Filing Separate, you can receive a child tax credit for the child that you claim.  Here is a source that outlines rather you would receive the full credit or not. The stipulation is that the child lived with you more than 1/2 of the year and if they got more than 1/2 support from you is the qualifying question that may be considered.  If the answers are no, she can still claim the child under the conditions of the separation agreement, but the credit would be reduced. She can also claim the childcare dependent expenses for the child she claims as a dependent, if the child stayed with her for more than 1/2 year. Please view this source about divorced parents living apart.
5. She can deduct the full amount of childcare expenses paid for the child she has custody of. 
6. Local back taxes paid are not tax deductible on a tax return. 

The problem I see is the 50/50 custody arrangement. To claim a full child tax credit and dependent care credit, the child would need to stay in a parents house for more than 1/2 a year. A 50/50 arrangement will not qualify and the credit is reduced to $500 for other dependent care credit whereas a child tax credit and dependent care credit would yield a larger credit for both spouses. This settlement arrangement may seem fair on the surface but might not fare favorably from a tax perspective for both spouses.

One more note. If the spouses file separately and if one spouse itemizes deductions, the other spouse must itemize even if the standard deduction is larger than the itemized deduction.

Publication 503            IRS Publication 501