First, Tasch to Zermatt is 12 minutes by train and there are around 83 trains per day. So obviously, not a factor in deciding which to stay in when the car must be parked in Tasch. Parking however may be a factor. It costs $16 USD per day to park in Tasch. If you stay in Tasch however, you park at your hotel for free.

Second, I never use third party booking sites for hotels in Switzerland (or anywhere else for that matter). Your comment that you didn't see a lot of choices outside of Zermatt is just one reason why, they don't list everything. I usually start any search on a given area of Switzerland on the My Switzerland website. https://www.myswitzerland.com/en-ca/taesch.html Then I might look for hotels in Zermatt, Tasch or Randa (a couple of miles farther from Zermatt but still 15 minutes by train). Randa is even smaller than Tasch. I might look for hotels with this Google map site: https://www.google.ca/search?sa=X&rl...6!2i490!4f13.1

Once I had picked a hotel I would then book directly with the hotel.

I don't know what you mean by 'nice' in terms of where to stay. Everyone has their own definition of that. We definitely prefer smaller villages rather than larger and less tourism rather than more. Zermatt is not our idea of a 'real' Swiss village by any means. It's as touristy as you can get. Tasch less so, Randa even less again.

As for where to hike, I don't know how familiar you are with how extensive the hiking trails are in Switzerland. Here is a link to let you see what to expect: https://map.schweizmobil.ch/?lang=en...1750&N=1189000 If you zoom in on the map you will see all the trails in any given area.

Lots of hike descriptions can be found on sites like this: https://www.zermatt.ch/en/Media/Plan...urs/Arigscheis