So many options!
I would recommend first reading these helpful pages:
https://www.seat61.com/european-train-travel.htm
https://www.seat61.com/how-to-use-a-eurail-pass.htm
Then don't hesitate to come back with more specific questions.
You do not need Eurail Passes, you can travel with standard tickets. No straight answer on which one is cheaper/more convenient. Seat reservations cost extra with Eurail passes, it's all explained above. ;)
For July (peak season, esp. in Italy) I'd recommend booking seats ahead whenever required. Not 6 months before (impossible), not on the day itself but perhaps rather 2-3 weeks ahead when your plan is well defined.
About your last question: if you mean border checks, they are totally random throughout the Schengen area. Border guards might walk through the train but you don't have to leave the train or anything similar.
(Not really important for your trip but I'll still mention it: Eurostar trains from/to London going under the Channel have a proper check-in procedure with security and passport controls as the UK isn't part of Schengen)
About destinations: too many options so I'll only mention my favourite railway.
The Bernina Railway between Italy and Switzerland is truly stunning in my opinion. There are hourly trains so it's really easy to travel on that line.
From Rome you'd go to Milan, then along Lake Como until you reach the Swiss border. Then the real fun begins haha. More details on https://www.seat61.com/trains-and-routes/bernina-express.htm
St. Moritz isn't the most beautiful place to stay but there are so many cute villages elsewhere in the Engadin valley, all accessible by train.
After the Bernina experience you could stay longer in Switzerland, travel to Munich, etc. Lots to think about!
