Should one apply for tenure track positions below current rank? Should a full professor apply for an associate professor position if it's the only way s/he might be able to continue his or her career? 

The situation in question involves a person recently promoted to full professor (after 5 years as associate prof. and 5 years as assistant before that, all at the same institution). At the time of application for the assistant/associate level position, the person will have held the rank of full professor for 1 year. The person's current position entails living 2500 miles apart from spouse and elderly parents for 8-9 months each year.
Contact the department/institution. If a department is hiring an "Assistant Professor", they have been granted a tenure-track slot (by the Dean or Provost depending on institution). In some cases they may be willing to hire at a higher rank than advertised if the applicant already has tenure elsewhere. Some advertisements are explicit in this offer, but the possibility may be there even if it is not explicitly mentioned.

Otherwise, the person needs to do what makes them happier. If moving closer to family is more important, then the person should apply. If the option to be hired at higher rank is not possible, then the person should probably indicate that he or she is willing to be hired at a lower rank/salary for personal reasons (however, it is unethical for the potential employer to ask what these might be).