All standard dishwashers need a 24" space. If the cabinet is a 24" cabinet then yes you can fit it. Wider will work with creative work but less than 24" won't for a standard DW

Most cabinets are boxes screwed to the wall and screwed to each other. And maybe some screws into counter top. As long as you have 24" then yes knock that box out and use a gable against the fridge for support. You need 24" clear is really all you need to know.