Is it OK to work on projects from research proposal before position starts?
I don't think you should ever delay work, especially in mathematics. It is like trying to turn off your brain. Work away. I'd even say publish all you want. You aren't being hired for specific pieces of work unless you are a consultant or such. But academic positions are nothing like that. 

The "worst" case is that you show up with everything completed and have to - horrors - work on something else that extends it or is otherwise "interesting." 

Kidding aside, nothing is off limits in advancing mathematics. You will only be praised for working in the interim. Your research proposal just assured the hiring group that you have ideas and will be productive.