Is it appropriate for me to claim my daughter and her children as tax dependents on my tax returns

The tax code has its own definition of whether or not your daughter and grandchild qualify as dependents. However, legally, if your daughter is a minor, you are responsible for her actions and may be sued if she causes damages or injuries. If she is an adult, your are not automatically considered a co-defendant for her actions unless some nexus to the circumstances exist, such as you let her drive your car and she has an accident causes damages or injuries to another. You should consult your homeowner's insurance, if you have such a policy, for its language regarding coverage as well. If you own property with her or her name is on your home deed/mortgage or if you have joint credit cards, you certainly may be liable for any debt for which she defaults. If you are considering evaluating a specific factual circumstance, you definitely should consult with an attorney for other areas of potential exposure.

Christina's answer was correct. As far as your filing status goes, I presume you already use the "head of household" designation, unless you are married filing jointly. If you have a choice between married filing jointly or head of household, it is usually beneficial to use the married filing jointly status.

Keep in mind that if you claim your daughter as a dependent, you must provide over half of her support. Also, if she is claimed as a dependent on your return, she cannot claim a personal exemption for herself on her return (since she is the dependent of another).