Gosh, these musical/political issues seem to be a lifelong issue for all of us. I have to say that in my present position they are not too bad, though one never knows how things might change when my pastor retires. I believe part of the resolution involves trying to find a place that "fits." I tend to like working in Catholic churches because I don't have to deal with committees--too many cooks in the kitchen. However, you have to have a good working relationship with your pastor or it can become difficult, to say the least. Some people thrive on working with committees, too, so everyone has to find what works best for them. I pretty much feel that I have been doing this for 33 years now (gosh that makes me seem old!) and have a master's degree from a well known music conservatory, so I consider myself to be a professional who knows what he is doing. The way I look at it is that I have about as much training, experience, and professionalism as a surgeon (though paid grossly less!) and I don't believe operations are done with a committee sitting around debating how it should be done.
On the other hand, I temper this with a huge amount of attention to people's reactions to the music I present. There are times when people have genuine gripes about what musicians do, and I try very hard not to be the typical "artiste" who won't listen to anybody. I always look for ways I can compromise without reducing what I consider to be the essential integrity of what I do. For instance, I have long ago stopped trying to win the battle on playing loud preludes. Since I work in a Catholic church where preludes are optional, I often do not play them at all, or only quiet ones. Only a couple times a year (Palm Sunday, for instance) do I play a loud prelude. I find it is very important to try to decipher what people are actually saying, as non musicians will often say things badly because they don't know how to describe what they're hearing. These amateur descriptions can at times be really offensive because they just don't know how to say it right.
On the other hand, there are times when people are simply obnoxious, and their issues are more about power, control, and inflexibility than carefully considered opinions. While I can be very patient with people, I am pretty impatient with these attitudes, which, unfortunately, are common in churches. I can recall one time telling a particularly stubborn choir member to go get a master's degree in music at Westminster Choir College and come back and then we'll discuss the issue.
As far as church politics go, I agree with Lars. I stay out of it as much as possible. When I was younger I would allow myself to get drawn into the petty backbiting and gossip, and worse, power struggles, and they ALWAYS backfired. I do not hang around the church, and I avoid the power struggles at all costs. It is tempting to try to bolster one's position through the back door politics, but it always created more stress and never worked for me. Basically, I am the "emperor has no clothes" type. I do not have closed eyes about all the ugliness that goes on in churches, but I try to live by my own moral code when it comes to others. What I find most difficult to deal with is the great number of people I see in churches who fervently profess all kinds of moral things, and then can't even seem to try to keep the basic "love thy neighbor" and "do unto others" ones. I try, as much as I can, to just remember these two and live by them and not worry about the other stuff. (While I like working in Catholic churches, I myself am not Catholic.)