On the first point yes, absolutely this is possible. When you select ‘yes’ How are you ‘transferring’ to the new process? I am assuming something like creating a new Case object that has the new process associated with it? In that case, you need to ensure the first stage of the process is something like “Completed by parent process” and after you have created the new object your next rule in sequence then completes that step - something like this is required to ensure that all the relationships to the parent case are in place before it starts execution. Now I may be over thinking this and it is just another block of actions in the same process…
If that does not give you any clues drop me a PM and more than happy to do a screen share to get to the bottom of it.
On your second point, I can think of two relatively easy ways to do what you want - and almost certainly there are others.
First think of it the other way round - When you click Yes jump to the next page - but fire an event to complete the stage - rather than clicking yes to complete the stage first.
Second, you could stay on the same page, but display different data based on the current stage name - depending on how different you want it to be!