I have read this thread with great interest and I totally understand and respect where BelArgUSA is coming from...
BUT... it is a VERY dangerous area that you go into regarding 'Troubleshooting and being creative'. (I'm not specifically talking about the fire test post engine fire for the B737 I'm just talking in general).
I cant but help think you are a very experienced pilot but there are many pilots with far less experience than yourself here reading these posts. Troubleshooting is one thing but not sticking to the 'Emergency Checklist' is something totally different and if for whatever reason right or wrong, you will have to explain your actions... I would not recommend it to anyone!
Granted many pilots don't have the greatest technical knowledge of their aeroplanes but the emergency checklist/aircraft procedures have been thought through by many people (designers, engineers, pilots etc) and there are reasons why you do certain actions and particularly why you do them in a specific order.
Doing things out of order or getting creative without the 'approval' of the airline and manufacturers procedures, well your asking for trouble which may ultimately get you fired or worse case lead to an accident.
At the end of the day the pilot is not there to trouble shoot (I say this with a pinch of salt) like a flight engineer would per say. Back in the day aircraft and there systems were different and had different levels of reliability... there were many cases of what you speak of 'Not in the books tips and tricks' to solve a situation/problem and thats why many of us always preferred having 3 pilots and having the experience and technical knowledge of the flight engineer with us.
Times have changed with the improved reliability of a/c, 2 man flight deck, better systems and maintenance along with better crm practices, pilot training. The whole industry is learning and getting better hence why air travel is so safe. We do as we are trained because we are trained well and thats why our industry is the safest.
To finish given a serious situation DO what your trained to do!! This does not mean you stop THINKING! Common sense is always good practice along with airmanship. The time to think about doing things differently to what you were trained or actions not in checklist or against them is ON THE GROUND outside of the aircraft, not in the air.
Happy and safe flying!
Aka.
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Experience is something you gain 3 seconds after you needed it!