Could a lawmaker, like senator John McCain, select someone to assume their duties while they are incapacitated?
The constitution and 17th amendment leave the specifics of dealing with Senate vacancies to the states. In the case of John McCain:
Arizona and Hawaii require the governor to fill Senate vacancies with a person affiliated with the same political party as the previous incumbent.
So, were he to resign or die, the answer is no: he wouldn't be the one making the appointment (he could suggest a name, of course).
Things become trickier in the event he's incapacitated, for instance by falling in a coma. According to this research here (emphasis mine):
There is no specific protocol, procedure, or authority set out in the United States Constitution, federal law, or congressional rule for the Senate (or the House) to recognize “incapacity” of a sitting Member and thereby declare a “vacancy” in such office. Under the general practice in the Senate (as well as in the House), a personal “incapacity” of a sitting Member has not generated proceedings to declare the seat vacant, and sitting Members of the Senate (and the House) who have become incapacitated, and who have not resigned, have generally served out their terms of office. In one instance in the House, a Member-elect who was incapacitated and comatose, and thus could not present herself to take the oath of office (Gladys Noon Spellman, of Maryland), was found not likely to recover, and the House proceeded to declare her seat vacant after the beginning of the new Congress. However, no such precedent exists for a sitting Member of either House who has taken the oath of office, and a vacancy with respect to such a sitting Member would generally exist only by virtue of resignation, death, acceptance of an incompatible office, or expulsion. Where “incapacity” of a sitting Member of Congress is concerned, there is no specific provision of the United States Constitution, of federal law, nor Rule of the Senate (or the House) that provides any particular procedure or designated practice. Clearly, when a Member of the Senate dies or resigns his or her office, a “vacancy” in the office is established that activates the procedures of the Seventeenth Amendment, that is, the “temporary appointment” of an interim Senator by the Governor of the State — when authorized by the State legislature — to occupy the office until a special or regularly scheduled statewide election is held to fill the term.
As to being able to say "this person will fill my boots for only a few weeks or months" it is, insofar I'm aware, not possible. Proxy voting is possible, but if I'm not mistaken it assumes that you give specific voting instructions for a specific vote, rather than giving what amounts to a blank check.