as has been said -- it depends entirely on the size of the parish/congregation. One fairly typical local (southern New England, USA) parish, for example, pays $16,000 per year. That covers two services per week plus holy days), choir rehearsals (one evening a week), consultation with the priest or pastor or whatever he or she is called, and availability for incidental services (funerals and weddings and the like -- although for those it is customary for the family to also pay the organist whether he or she plays or not, incidentally, and sometimes they actually do...). A more ambitious parish might well require more, such as office hours or more choir rehearsals. It is not that common, however, to find a position which is what most of the world would consider "full time" with full time pay, although they do exist -- mostly in cities. (the Minister of Music of a cathedral with a choir school -- a most enviable position! -- is most assuredly full time; indeed if anything overtime -- and is paid accordingly. Indeed, rather handsomely, as ecclesiastic positions go!)
I would add, however, that even if the pay isn't full time, you should expect to spend at least 20 hours a week on a job such as the parish I described above, including your own practicing. The hours, however, except for the services and rehearsals, are quite flexible, of course -- and you would probably be practicing that much anyway! Some parishes -- not all, by any means -- allow taking students.
(A disclaimer before I continue: I was a church organist or minister of music for almost 50 years; never full time. And loved almost every minute of it!)
As birba noted, it definitely helps if you are a good sight reader. In fact, I would say that that is almost mandatory (sight reading makes up almost a quarter of the Associate exam for both the RCO and the AGO). Improvisation is somewhat helpful, but not really mandatory. Having available a wide repertoire in various styles and eras is, however, and it must be kept current. It also helps if you are at least moderately familiar with the liturgy or liturgical style of the church you are planning to play for and to be sympathetic to it even if you are not a deeply committed believer. If you have a choir, a familiarity with a wide range of choral music -- again, various styles -- is helpful, as is in most cases a pretty high level of tolerance for somewhat untrained enthusiasm -- and you will need to know how to conduct from the keyboard.
Organ technique and piano technique are not the same. In the type of parish I described just above, you can get by with good piano technique; it's not that no one will notice, but that it is accepted (trained organists are rather thin on the ground). In a bigger parish, where you might well expect to get a living wage, you will need to be fully versed in organ technique and literature.
There really is no other job quite like it. I loved it. As trus said, the acoustics of some old cathedrals are fantastic -- never mind the instruments available (although I can assure you -- the first time you open up a really big organ in a space like that it is a very humbling experience) and some of the most wonderful music ever written is sacred.
Topic: Becoming a Church Musician