Rant alert (again!)
I have no one but myself to blame for my current predicament, so I'm not asking for any sympathy, but I have been seriously wondering about this question.
Firstly, I fully accept that there are some top organists' jobs with high-flying organists in them, all employed on a properly professional basis. I'm sure they have their problems, but I'm not really concerned with them. I am wondering mainly about your average professional and semi-professional organist trying to scrape an income at a more humble level and how they might compare with other musicians of similar standing.
As far as teaching goes, I can't say a great deal as I only do a miniscule amount and I know I have it easy. My impression, however, is that organists are in much the same boat as any other instrumental teacher, sharing similar rewards and frustrations - but that, I suspect, is where any similarity ends.
If you're an organist you may have a church job. If you are exceptionally lucky it will be somewhere where music is still wanted, valued and appreciated. However, the overwhelming chances are that all your congregation really wants is a human karaoke machine. They're not really interested in music. You could be Carlo Curley reincarnated and still no one will bother to listen to your voluntaries. While people are taking communion you may treat your hushed audience to some carefully chosen organ repertoire, but, actually, most people would much rather you just played hymn tunes for them to hum piously. You will be lucky if you are paid a realistic fee for your skills; indeed, some churches (the free churches especially) will expect you to donate your expertise as a freewill offering. Weddings and funerals may be better remunerated (though a frankly derisory sum of just £20 has recently been cited in another thread), but mention copyright surcharges and you will probably be thought uppity. If a neighbouring church asks you to give a recital, don't expect to be offered a fee.
Since quitting my last church 18 months ago I have found myself in increasing demand as an accompanist for local chamber and community choirs. This might sound good, but actually it's simply because there's no one else stupid enough to do it. What's the problem? Well, basically it's the things you are required to play. If you ever get given something with an original, genuine organ accompaniment count yourself blessed. Most of the time you are substituting for an orchestra, playing from a part that is, at best, arranged for piano (left hand in octaves, right hand somewhere off the top of the keyboard) or, at worst, a literal reduction of the orchestral parts without any thought for performance practicalities at all. Recently I have noticed that the programmes are becoming progressively more problematic. My last concert featured Brucker's E minor mass. Its original wind band accompaniment sounds simple enough on recordings, but the interweaving counterpoint reduces to a keyboard part of sometimes quite horrid intricacy and the endlessly modulating chromaticism doesn't help either: the Duruflé Requiem was, literally, a lot less trouble. My next gig is a complete Messiah (Watkins Shaw edition). What on earth possessed me to agree to this one? The keyboard part, in which the orchestral parts are combined with a continuo realisation, isn't even playable by two hands much of the time. As always, I have to re-arrange the score as I play and try to commit my solutions to memory. For a year now my whole musical life has been almost totally dedicated to learning accompaniments - all for a paltry remuneration which, although the choirs concerned would no doubt call it handsome, actually amounts to slave labour for the hours involved. I wonder what a proper professional organist would charge for this sort of work and, indeed, whether a real professional would waste his time on such things. Maybe my musicianship is at fault for not being able to sight-read these things note-perfectly and take them in my stride, but I'm not sure I would believe anyone who claimed that Messiah is easy to play on a keyboard at modern speeds.
Having seen the way things are going I resolved to decline any further gigs where there was likely to be a lot of work involved. That hasn't worked either. Two concerts I have accepted for 2015 have already had their programmes changed! Fortunately the substituted pieces are all within my repertoire, so I'm not pulling out - but it does seem that, once you are booked, you are regarded as fair game for anything.
Go on: tell me other instrumentalists have it worse.