I would not agree with your prior preparer and here is why.
In order to qualify for the head of household status you have to meet a number of criteria. One of which is that you can claim an exemption for a dependent and your home was the main home of your child, stepchild, or foster child for more than half the year. Since your child can only live with one of you for more than half the year only one of you will qualify for the head of household status.
My answer would be different if you had two children and one child lived with you for more than half the year and the other lived with your spouse. Then you both could attain the head of household status.
See link here to Publication 501 regarding the Head of Household filing status. - http://www.irs.gov/publications/p501/ar02.html#en_US_2011_publink1000220775
I hope I have provided the clarity you were looking for, please let me know if you have any further questions.
However, you stated that if each had a child at one of the homes you would feel differently. If the child is living with the one with the home, can she claim head of household. - If she meets all the other requirements then yes she could claim head of household. If so, what would the filing status of the other spouse. If she can claim head of household and he can't and they are legally married, what does he claim--Married filing separately or single. - he would be filing as married filing separate. He cannot file single as he is legally married. Also, if the status' are separated with the separate status, do community property laws come into affect since Texas is a community property state. - Yes they do. Basically you have to claim 50% of all community income (i.e. joint income) and all of your separate income. There is a nice chart on page 3 of publication 555 which outlines this. See it here http://www.irs.gov/pub/irs-pdf/p555.pdf
What I am seeking is your advice on the appropriate status for this year under these circumstances for both parties. If there is a head of household option for one; what is the appropriate filing status for the other. - If the children lived with one parent for greater than 50% of the year then that parent could claim HH status and the other parent would claim MFS. If one child lived with the other parent for more than 50% of the year then both parents could claim head of household.
And if that is possible, is income treated as community property and added and divided equally with the separate status OR should they file either as MFS or MFJ. - See my comments above regarding community property income.
I hope this provides the clarity you were looking for, please let me know if you have any further questions.