Hi Lindsay,
I'm not sure if you are using AGOL or Enterprise, but you can only see some of the fields in the templates tab of Field Maps web app because of a known bug. There a couple of posts floating around discussing just that.
In order to see a field right now it needs to have a default value assigned. In order to assign this default value (on a field that doesn't currently have one) then you need to be in map viewer classic.
You don't need to worry about messing up your map viewer config so long as you don't save the map in map viewer classic. Updating feature templates does NOT require you to save the map... it has its own save operation. But, to avoid any map viewer configuration issues (which should only be a problem in Enterprise now) or accidents don't open your configured map in map viewer classic at all; just open the layer/feature service in a new/stand-alone map viewer classic and mess with your templates there, then there is nothing to mess up.
Once you have updated your templates in MVC, then those fields will show up in the FM Web app templates.
As far as your other question is concerned, I don't think you can have separate templates for the master layer vs the view (not 100%). You either need to create all the templates in the master layer, including what you want in the view or go with dynamic default values using field calculations, edibility constraints, and probably some helper field or config your map to achieve what you want. This will be much harder to do in Enterprise at the moment (but possible) and is still a bit hacky in AGOL.
- Justin Reynolds, PE
