Your questions are somewhat tricky... It all depents. You need to look up Publication 501.
http://www.irs.gov/pub/irs-pdf/p501.pdf

There are three different issues here ---

1. Head of household
   A taxpayer qualifies for HoH status if BOTH of the following exist:
 (1)unmarried ( or considered unmarried) as of Dec 31,
 (2)you paid more than half of the cost of keeping a home that was the principal place of residence of a dependent.  ( your dependent parent doesn't need to live with you. but you paid half of the cost of keeping his/her home.)

2.Dependent
   In order to claim a dependency exemption, the dependent need to meet several tests:
   see Publication 501 Page 10 , table 5.
   eg:  the parent must be citizen/resident , gross income must less than $3,300.

3. Married Filing Separately.
 It is better to avoid MFS. The reason? See P6. "Special Rules."
 eg: When your spouse has mortgage payment or for whatever reason needs to file itemized deductions, MFS won't allow you to have standard deduction. Your MFS return needs to itemized deductions too... You have to pay more tax when you don't have anything to itemized...
 If you are living in one of the community property states, it causes more trouble.