The problem with your questions Bob, is that they are rather general. As such, "specific situation" answers are quite difficult. You might well get 6 different answers from 6 different people and all of them might be right in a given set of circumstances. Obviously there are guidelines, checklists and common sense applications that will have a large bearing on the decision process, however it is often the case that a commander and (his) crew will have to draw information from numerous sources and adapt a number of solution guides to resolve any given situation.
Fire that you cant extinguish
If the nearest suitable airport has weather below your minimums (say 100 feet lower, airport not cat 3). Would you go below you minimums or choose an airport that is a little bit further away with good weather
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For a start the airport would not by definition be suitable ? The cloudbase is in any event not relevant to certain types of precision approach (cat 3 or not). Obviously any fire that cannot be positively extinguished is a very serious event. In such circumstances a crew would be aiming to land the aircraft as soon as possible, since any delay might remove the ability to make that choice. However an airport with good weather that is "a little bit further away", (little bit ?) would likely result in an earlier landing if a visual approach negated the need for an instrument approach to an airport with poor weather conditions. It really is a case of "How long is a piece of string"? On the other hand a fire that occured to an aircraft already on an instrument approach would have to be weighed up against the additional time delay incurred in diverting and in such circumstances the crew might well elect to fly below the published minima in regard to the risk involved.
Uncommanded rudder /Jammed flight controls/Runaway stabalizer
If you managed to resolve any of the above problems would you continue the flight or make a precautionary landing
Again it all depends on the reason why the problem occured. Certainly if there was no obvious and rectifiable cause then a precautionary landing might well be carried out. However there are situations that might cause or result in some of these conditions that are identifiable or resolvable that the crew might judge as not warranting a diversion. For example freezing water can sometimes cause jammed flight controls, and a descent into warmer air can resolve the problem. A rudder or stabilizer movement (runaway) might be caused by an erroneous pilot input.
At the end of the day not everything is as black and white as it might seem to the casual observer. There is plenty of guidance that together with training and experience means that a problem can often be approached from different directions depending on the information that is available and the time criticality.
Airmanship is in part about the ability to utilize common sense, experience and resource within a team framework to bring about the safest possible outcome for any given task or operation. Often that combination will result in the utilization of routine procedures and checks with (when required) the use of emergency procedures and checks. However that does not exclude the use of reasoning and even inventiveness when required.
Seeing the broader picture and how a problem is dealt with within it, and ensuring that we know how to utilize the resources that are available to us in our roles as crewmembers is why we are (or should be) paid the big bucks.