Murren, Switzerland to Florence,Italy

It is more a case that for Murren to Spiez to Milan that you would save money buying that in advance instead of the day of. https://www.sbb.ch/en/timetable.html
When you buy the Italian rail leg it will come with a reserved seat assignment.

http://www.trenitalia.com/

You never need reservations for Swiss trains.
You always need reservations for Italian long distance and High speed trains. Different companies, different rules. And such Italian trains are cheaper if bought in advance. And Italian tickets on these trains are specific to one train and include a seat reservation.

The route is: Mürren - Interlaken - Spiez - Brig - Domodossola - Milan - Florence. But you don't need to know that in advance, just enter Mürren to Florence in the SBB website ( https://www.sbb.ch/en ) and it will show you the options and route.
When buying tickets what you do need to know is where you change on to the cross border train.

• Sometimes it is Spiez, you get one train Spiez via Brig and Domodossola to Milan. In which case you buy Swiss tickets Mürren to Spiez, and buy on the Italian Railways site ( http://www.trenitalia.com/tcom-en ) Spiez to Florence.
• Sometimes it is Brig, you get one train Brig to Milan. In which case you buy Swiss tickets Mürren to Brig, and buy on the Italian Railways site Brig to Florence.
• the third option is if you have a Swiss Pass. This is valid up to Domodossola. In this case all you need to buy is Domodossola to Florence on the Italian Railways site.

I am seeing some weird routing on SBB starting June 10 and continuing well into July.

Morning trains fron Interlaken Öst go via Luzern, Arth-Goldau and the Gotthard tunnel, instead of the faster Simplon route via Spiez and Brig. Only after 13:00 or so does the Brig route show up.

Are they working on the Simplon tunnel?

The EC trains from Basel and Geneva run the route in the early morning and late afternoon. Between them, it is connections to Regionali in Domodossola. The Regionali are still not scheduled after June 10, so that is why you don't see anything late morning to late afternoon.

This is your most convenient option on the current timetable ...

RE 148
Dep 13:02 LAUTERBRUNNEN (Switzerland)
Arr 13:23 INTERLAKEN OST (Switzerland)
RE 6723
Dep 13:29 INTERLAKEN OST (Switzerland)
Arr 13:50 SPIEZ (Switzerland)
EC 57
Dep 14:05 SPIEZ (Switzerland)
Arr 16:37 MILANO CENTRALE (Italy)
FR 9549
Dep 17:20 MILANO CENTRALE (Italy)
Arr 18:59 FIRENZE S.M.N. (Italy)

Thank you so very much for all of your wonderful advice!
Best regards,
Viki