You should use your address on their return if that is where you want any IRS correspondence to be mailed. Just be sure to prepare their State return based on where they actually live, not their mailing address.
You can sign the form on their behalf if you have Power of Attorney. There is an IRS form you can use if you don't already have a general power of attorney: https://www.irs.gov/taxtopics/tc311.
