What you're talking about is creating a 'methodology', and yes, it's a good idea.
The methodology is created for the reason you mentioned - so that (most) anyone can step in and keep the project moving, because it follows an established process and requirements.
As for what to include - that depends on the project type and what your company feels is important. At the very least you will probably want a project charter or scope statement (what it is we're doing, what's included, who's the client, etc.), an estimate and budget procedure (how do we develop the estimate, how do we define the budget), the processes for execution of the project, the monitoring processes, and the close out procedure.
The Monitoring process addresses the improvement and measurement aspect of your question. It's here that you'll determine what's important and needs to be monitored/evaluation, how often, what metrics, what variance tolerances are acceptable, etc.