When you store it as a single string, which order will you choose? Will you have delimiters between them? Would they still be in effect when the person's name has more than 2 parts? If you are ever in a situation where you need to sort by name, will you do it by first or last name? What if there is a need for the other option to the one you chose? Can you reasonably guarantee that you can identify family members if you ever need to? When the customer tells their name over the phone, will it slow down your customer service people if you display the entire name in a way which they are not usually familiar with?
2. Sort by the whole thing
3. No you can't guarantee identifying family members any more than with a first name last name system. If you need that it should be an additional part of the system. Presumably for the 90% of people who does have a typical first name, last name combo you can still identify family members in the same way.
4. Just introduce free text search for this field if you need to find people.
But suppose you are working at a place where people are keen on getting 'smart with the data', your replies do increase the entropy if only to the extent that 'we have this idiot programmer who thinks he is too smart to use first and last names as we have always done and is breaking all our reporting systems and is causing hell for our customer service agents'.
:-)