Can I use an index to optimize select clause, versus where clause?
Query 1:

INDEX(person_type, person_id) -- in that order
INDEX(person_type, person_id, full_name) -- to be "covering"


Query 2 ("covering" is not practical because of *):

INDEX(person_type, full_name) -- in that order


Query 3:  Before making suggestions here, please explain why you have a LIMIT without an ORDER BY.

See also my cookbook on making indexes.