If you took care of family members out of state, you could add that to your "Work Experience" section of your resume.
Something like this:
Caregiver (November 2018 - December 2018). City, State
Provided in-home care to elderly family members. Responsibilities included medication set-up, meal preparation, and transportation to and from medical appointments.
For hurricane disaster relief, if that was provided on a voluntary basis, you would want to add a section to your resume titled "Volunteer Work" and describe the work the same way as above.
I'd recommend leaving off all the other activities, including the care you provided to your dog [I'm very sorry about his/her illness ].