Lots of good ideas. You are very wise to spend several days in each of your bases. You have avoided the disease of trying to see too many things in too few days. I do have some concerns about your choice of bases.
1.- Car rentals.
Automatic transmissions are optional, although they have become more common. Manual transmissions are the norm on rental cars.
If he cannot find a sports car, consider a VW Golf. Excellent handling. You should note that Swiss speed limits are rigorously radar-controlled at many, many places, and that speeding fines for even modest infractions can exceed 1oo CHF or even 200 CHF. If you are careless, your rental car company will send you a notice that the police would like you to pay a very large bill... If you do not pay , don't plan to go back to Switzerland. Their computers are very alert.
I rent and drive with a US license. An international license is not needed.
This map may help choose roads:
This extremely detailed and flexible topographic map can give altitudes: map.geo.admin.ch/?topic=ech&lang=en&bgLa yer=ch.swisstopo.pixel karte-farbe&layers=ch.bav.ha ltestellen-oev,ch.swisstopo.swiss tlm3d-wanderwege&layers_visi bility=false,true&X=16 3750.00&Y=633225.00&zo om=4
I have turned on "Hiking Trails."
2.- Weather and locations.
Late May and early June are not yet Spring at the higher altitudes at or above 2000 meters, such as near the Jungfrau. Most high altitude hiking trails will be closed by residual snow that is not very good for skiing. So, no skiers, no hikers, no customers, and many hotels and other services are closed and do not reopen and begin to serve the hiking season until mid- June. A good deal of the transport services do open on about May 25. There are still good things to do, but it is truly "low season." Trails at lower altitudes, such as on the valley floor, will be clear. See attached image.
Weather is highly dependent on altitude. Most high Alpine road passes do not open until early June, as a rule of thumb. There are exceptions, depending on snow pack.
Zürich - Most first timers do not find that Zürich holds their attention for 4 days. If you have specific activities in mind, then you should, of course, spend the time there. If you are interested in cultural activities, and, even better, if you speak or read German, there are many ways to explore cultural and artistic venues. English signage is common in places such as museums, but not universal by any means.
Generally, first time visitors find that Luzern has more variety, more scenery, and more tourist-focused activities or scenic places.
If you use the timetable, you will see that Luzern and Zürich are only 45 to 50 minutes apart, twice per hour, by quite nice trains.
The timetable will help you plan all your travel.
I'd stay in Luzern and commute to Zürich as needed.
By late May, Spring is well advanced at the altitude of Luzern (and Zürich).