Village of Zweisimmen and Zurich airport hotel

On that route you also change trains at Interlaken and usually Spiez.
I would not call Interlaken a small town. However if you head south from Interlaken for 30 minutes (by train) you get to Lauterbrunnen.
Lauterbrunnen looks like this (click for photo).
More info and hotels here: https://lauterbrunnen.swiss/en/summer/

"looking for a decent, but low-priced, hotel close to the Zurich airport and within walking distance from that hotel to the train station that would allow us to travel to Zurich city center." - but why do you want to see Zürich city centre?
If you want to see Zürich city centre, stay there, it is only 10 minutes from there by train to Zürich airport. However I would recommend spending your last night in Luzern. It is only 70 minutes by train from Luzern to Zürich airport (first train 04:55, arriving 06:11, so no problem with early flights).
Luzern tourist site (info and hotels) here: https://www.luzern.com/en/

"A hotel with a . . . a free shuttle to the airport would be nice." - Switzerland doesn't do shuttles, it does trains. The only hotels with shuttles are those near the airport but not on any public transport to the airport.

Thanks. We will be in Lauterbrunnen/Murren earlier, so I am definitely thinking of an overnight stop in Zweisimmen to break up the trip from Chamonix/Montreux to Luzern. What kind of a place is Zweisimmen? Pretty? Hiking trails? Authentic? I may revise my plans depending on what people tell me about Zweisimmen.

Regarding the question about the Zurich airport hotels, we will have been in Luzern earlier in our travels, also, so no need to go back there. Actually, we will be coming from Bern on our way to the Zurich airport. But we have an early flight (7:40 am). Should we be at the airport two hours early (5:40 am)? If so, would there be enough time to stay in Bern the night before the early flight and then train to Zurich airport with plenty of time to catch our flight? So, if that works out, we could stay an extra night in Bern. Does that sound like a good plan?

We are still open for suggestions on a decent, but low-priced, hotel close to the Zurich airport and within walking distance from that hotel to the train station that would allow us to travel to Zurich city center. A hotel with a very early breakfast (5am) and a free shuttle from the hotel to the airport terminal would be nice.

The first train from Bern to Zürich airport in the morning is 04:21, arrive 05:39. How long before departure you need to be at the airport depends on where you are flying to and which airline, could be 1, 2 or 3 hours. I don't think staying in Bern and catching aa 07:40 flight is practical.

"low-priced, hotel close to the Zurich airport and within walking distance from that hotel to the train station that would allow us to travel to Zurich city center." - I think you are looking for the impossible here. Any hotel at the airport would be expensive. If it is near the airport, you would need to get a bus both to the airport and to a station.
For a flight that early I would pay for a hotel at the airport, you would then be in walking distance to both the terminal and the airport station. The only such hotel is the Radisson Blu: https://www.radissonblu.com/en/hotel-zurichairport

You can find hotels near the airport, with a shuttle, by entering the search terms "Zurich airport hotels"" on TripAdvisor. Here are the first two listed after the #1 rated Radisson Blu, which Chris F mentioned. ( We have stayed at the Radisson Blu and it is very nice, and very convenient as it is actually in the airport terminal, but quite expensive).

This one offers shuttle service to the airport starting at 5 am but you must reserve seats:

https://www.parkinn.com/airporthotel-zurich?facilitator=REZIDORTRIPADVISORBUSINESSLISTINGS&csref=aff_ta_pd_cl_ZHRP1&dclid=CPDZydX_qeACFQHNZAodwGMFBg

This one says they start serving breakfast at 4 am and the shuttle starts at 5:

https://www.ihg.com/holidayinnexpress/hotels/us/en/zurich/zrhap/hoteldetail?newRedirect=true&qIta=99618083&icdv=99618083&glat=MEDI_mdpr-_-TripAdvisorUS-_-BLwebsiteD-_-ex_zrhap&qSlH=zrhap&qRms=1&qAdlt=2&qChld=0&qCiD=14&qCiMy=042019&qCoD=17&qCoMy=042019&setPMCookies=true&qDest=Hofwisenstrasse%2030,%20R?mlang,%20CH&cm_mmc=mdpr-_-TripAdvisorUS-_-BLwebsiteD-_-ex_zrhap&srb_u=1

However, like Chris F I would advise you to stay in Zurich itself, near the train station, if you want to spend the evening exploring Zurich. It is only a 12-minute ride to the airport, and the train starts running early. If you cannot find a place with an early breakfast, there are plenty of options for that in the airport.

As for Zweisimmen, we have only been through there on the train one time, not stopping over, so I do not remember much about it. But it is a very pretty area, with ski and walking trails nearby. You could take a look on Google Maps satellite view, and then walk through the town with street view. The map shows a hotel very close to the train station, named Gasthof Derby. It sits near the base of the Rinderberg ski lift. it looks like a very traditional Swiss mountain inn and gets good reviews on booking.com. Note that there are no rooms with ensuite bath; there is a sink in each room but the toilet and shower facilities are shared.

http://www.derby-zweisimmen.ch/index.php

You can get more information on the village of Zweisimmen, and find more lodging options, from a tourism website such as this:

https://www.myswitzerland.com/en-us/zweisimmen.html

Note that they describe it as "quaint", which is probably what you are looking for.

Here's is a possible hike that starts at the Zweisimmen train station:

https://www.gstaad.ch/en/summer/hiking/cultural-and-themed-trails/detail/zweisimmen-castle-trail-circular-route.html

The Tourist Information Center in the village should have brochures on this and other hike suggestions.