If you get a second bunny you will have to wait until 1-2 months after both alterations. So if you get a baby you have months to wait before you can put them together. So yes, they will need separate everything and should remain separate at all times, including play time. Bringing a new bun into the house won’t make him lose his litter habits. When you start the bonding process you’ll probably see territorial poos and some marking.
After both procedures have been done, while you’re waiting the 1-2 months you do what is called prebonding. Switching their items between their cages and switching them between the cages. When the 1-2 months is over you can begin bonding sessions in neutral territory. Start out with short sessions, and the more friendly they get the more you can increase the time. The ultimate goal is for them to spend a full 24 hours without any squabbling. You need to supervise them during all bonding sessions.
Bonding can be a very long process, and it is very involved.

Absolutely! That’s one of the best things you can do for them. The cages should be at least six inches apart. It helps so much that they can be able to see each other and interact through the bars, especially during play time.