Welcome to my Swiss Alps-
Sounds like a great trip.
I'm not going to be able to answer all your questions/interests at this time, but here is a start.
<<"I want to explore every part of Switzerlands country side-- if that is possible. I would like to do a lot of hiking ( no overnight hiking because I will have luggage that I intended to to keep at whatever hostel I stay at).">>
There is good luggage transport system on the trains, which might help.
The timetable will help your planning and travel:
This map is god on transport. Zoom in and play with the menus:
This website can help with details of many hikes. You me need to spend some time with it, but once you find hikes...and they include the best...you can get altitude profiles and accommodation details.
Here is one example:
<<"I want to explore every part of Switzerlands country side-- if that is possible.">>
To explore every part of Switzerland's countryside will require approximately 123.7 days, and require a visit during the peak of the hiking season.
Just joking...it would actually require more than that. ;-)
That is July 1 through September 30. You will note that that is only approximately 90 days. Thus, you will adjust your hikes to lower altitudes at the beginning or tail ends of that time, or comeback again.
This splenid and exceptionally scenic, very easy short hike might be open in early June, with luck and a melt off of much of the snowpack that can slide and block the trail. June 15th is the usual approximation for the opening date. Starts at roughly 2200 meters and finishes at roughly 1900 meters....if my memory is working tonight.
This really good but not easy one will not be:
<<"I will be in the country early June.">>
By choosing that time, you have excluded the best high Alpine hiking. It is still the tail end of winter at 2000 meters and above. The SAC huts, which experienced Alpinists use, tend to open in late June or early July.
After you have taken a look at the links on the home page, such as-
and begun to focus on possible ideas for where to go in your very limited time, come back with some thought about your itinerary. We can go from there.