These are good and complicated questions, I'll try to answer them as best I can but feel free to ask more if I confuse things in some way, but want to state the obvious that just like in real estate, EVERYTHING is negotiable in construction.
1) Are you using a standard form contract? AIA? Which one? Most of my projects are GMP (guaranteed maximum price) and run in an open-book format which gives us a lot more flexibility to tweak things.
2) What size of projects are you doing and how many at a time?
3) Are there specific trades you have issues with in finding skilled labor? You indicated finish carpenters, is that all?
If I were you and that was my weak link / main constraint I would direct hire some guys to do that part and treat them very well. This can be excluded from your GC's scope, but be specific and the coordination will take some extra teamwork (i.e. maturity). Note - insurance and liability will also need to be addressed.
If you can't justify full time labor, you could also look to hire install crews out of a local cabinet shop. When I was a cabinet maker I wound up doing finish work frequently. I was more expensive, but much, much better than any guy off the street.
You could also hire local (good) guys to do it piece-work on their weekends. Often the best quality guys are also among the fastest - they are just better. When I was a carpenter I made great extra cash doing this. Me and another guy could hang doors, cabinets, and all trim in a house in one day.
If these items are also your contractors point of frustration they will HAPPILY let you handle that scope.
4) Yes, finding new GC's is rough. Do due diligence as if this was your first project ever by looking at other projects, taking to past clients, etc. Ask the clients open ended questions and let them talk. Also make sure to ask your contractor to provide a reference of someone who doesn't like him - they will obviously resist and you will have to assure them that we all have someone who doesn't like us, justified or not. You are doing this with everyone and will take everything they say with a grain of salt.
I also recommend you ask to see an example invoice, PCO, etc. That way you can make sure it meshes with your internal system before the headaches start.
I haven't done this, but I imagine you could also flip the standard on it's head and have the builder tour one of your finished products so you can discuss expectations in detail. Take lots of photos and attach them as an addenda to the contract. That way come punch-list time there are no arguments.
THE KEY TO DOING THIS WELL is to have all of your arguments ahead of time. Then it will go smooooth when it counts (*knocks on wood).
5) I rarely use incentives but it really depends on the situation. How much is it worth to you if they finish early? How much does it cost you if they are late? It may not even be worth discussing since a smart contractor makes money is by finishing a project as fast as possible. Just like your business, the faster you can cycle through deals the better your margins, and the more deals you can do.
The only time I think this is really important is if you have a hard deadline like a commercial lease with liquidated damages, a balloon payment on a loan, or something similar.
Budget based incentives really depend on your structure. If it's open book you can do a shared contingency where the GC gets a % of unused contingency. However, they will then be incentivized to classify things as change orders instead of absorbing them in their scope. Always remember that people do what you incentivize them to do.
6) Rain delays and other unforseen items are always the cost of the owner. How a delay is defined and how much it costs needs to be established before hand (see note on having all your arguments ahead of time). It is also important to establish if it actually impacts critical path - if it rains and they can't pour the driveway, but the interior flooring is critical path, who cares.
How you do this, again, depends on your contract structure. I typically require a GC to carry 7-18 delay days in their schedule depending on the project, but my projects are typically open book and if the delay isn't used then it isn't paid for.
The prenegotiated delay day cost should just cover the basic overhead expenses and nothing else trailer rental, toilets, insurance, etc).
Long winded, and I'm sure raised just as many questions as it answered. Feel free to ask more.
SB