Your question is really a lot of questions, so not ideal (and really not allowed on this site). In any case some answers:
Swiss people tend not to use taxi. You can do most things without taxi (but if you are busy businessmen always at phone).
Then it depends on you. On train all group see much more: better view and no driver. You can also walk and discuss, point things and plan. But you trade flexibility. Trains (and busses) are good to link localities (so places with inhabitants or workers). It is far less convenient to spot few natural places, or to do some alpine passes (infrequent busses, so you cannot stop and explore, but maybe at summit).
Interlaken, Lucerne, Engelberg are easy to reach with trains. But if you want to see more spectacular parts (Lauterbrunnen valley, etc.) train/bus are not ideal.
Personally I think Piz Gloria (Schiltorn) is much better and less touristy and cheaper. Possibly less thing to do on top, but you are there for the mountains, so... (no walking on ice, etc. just a 007 exposition). Or Zermat (which it is touristic and nice). Engelberg: I'm impressed by the numbers of busses in there, and not much more.
About weather. It is far from your arrival, so nobody know. At that time, you may see first snow on mountains. Titlis and Jungfrauhoch have better entertainment inside buildings, and you get the spirit of mountain: you may see the sky and 3 minute later you cannot see your hands. Some magical environment we read in books. But nothing more. I'm not sure it is worth the travel (OTOH once in lifetime may be interesting). If you have flexibility, plan to go on top of mountains when forecast is good (and stable).
Also parking is very expensive in all Swiss cities (and hotels in cities).
Depending on the numbers of day in Switzerland, try to do both.