I'm new to the logic we can use when forming a condition for a rule.
It seems that whenever I add the && it doesn't work, and it causes my other rules on the form to stop working.
How would I make the following rule work?
You could try writing your rule like:
and(is edit mode,equals(my requester approver(display name),Current User(display name)))
Also the second rule will not work. The not() runtime function is designed to reverse the outcome of a rule in Nintex Forms. It cannot be nested inside other functions.