1. Beside a valid ticket for your journey, you do not need a reservation, this is not mandatory. You can make a seat reservation here: https://www.sbb.ch/en/travelcards-and-tickets/tickets-for-switzerland/seat-reservations.html
2. Your tickets should be visible in your account if you have logged in before buying the tickets. If you have not been signed in - for what reason - the tickets cannot be added to your account afterwards.
3. 6 minutes should be enough time, because the plattforms are near each other. If you do not have a super save ticket (if the train number is not written on the ticket), you also can take a train later if you miss the original train.
And a pro tip: Look for the “IC81” train in the schedule on sbb.ch or in the SBB App. This connection IC81 operates directly between Zurich (and Zurich airport) and Interlaken, for example here. But then please verify that you do not have fixed super saver tickets only valid for a specific train as I mentioned above.
4. You do not have to check your bag in. You can take your luggage with you. There should be enough storage space available
I wish you a nice stay in Switzerland!