Aperture Priority 80% of the time as for some reason when using flash, the camera doesn't like Ap and slows the shutter to ridiculous speeds so i put it into P.
5D | 24-70L | 28mm f/1.8 | 580EX
Mainly Aperture, sometimes Manual.
For 1), I use manual when I don't expect automatic exposure modes to work well - e.g. for high contrast scenes, backlit scenes, etc. For simple situations automatic is usually good enough.
Exposure correction and exposure lock features (possibly in combination with partial or spot metering), if easy to use, also reduce the need for manual mode. By manually picking the scene/object on which automatic exposure is based, or by manually applying an estimated correction factor, one ends up with semi-manual exposure. (However, this doesn't work well with "intelligent" matrix metering systems which can be quite unpredictable).
Concerning 2), one has to realize that ALL automatic modes are "program" modes. The "program" is a curve through the 2D grid of exposure times and apertures (or 3D for exposure time, aperture, and sensitivity). For shutter priority, the "program" is a straight line parallel to the aperture axis; for aperture priority, the line is parallel to the exposure time axis. For "P" mode, the curve depends on the camera. (Some cameras, e.g. from Minolta, also allow[ed] do download custom program curves.)
If the desire is to e.g. get a specific exposure time, one can achieve this in all 3 modes: by directly selecting the desired time in shutter priority mode, by setting an aperture that results in the desired time in aperture priority mode, or by shifting the "program" curve until one gets the desired time. It works the same if you want a specific aperture. On top of that, with many contemporary cameras, all three methods have exactly the same user interface to achieve this - spinning a little wheel. So the operation and the result will be exactly the same, one just starts from different initial aperture/time combinations.
The only real difference arises is if you want to maintain a specific aperture or shutter speed when the light changes - aperture or shutter priority then save the time to spin the little wheel to readjust. In most cases, I'd dare say it doesn't really matter.
For snapshots, "P" mode is usually the best bet. Shutter priority gives only a relatively small window of aperture values which can be varied to achieve the necessary exposure; when you hit the limitations (i.e. you require 1/500s exposure in a dark room, or 1/30s in bright sunlight), there's nothing the camera can do to save the exposure (although some cameras might automatically switch over to "P" mode behaviour). In aperture priority mode, the range of available exposure times is much wider, but not if you require a short enough time for hand-held shots. "P" mode can vary both time and aperture and thus has much better chances of delivering well-exposed, non-blurred hand-held images with reasonably stepped down apertures.
Conclusion: If you have the time to make concious decisions, the various modes will all let you achieve exactly the same result (it takes on the order of 1 second with typical cameras). If you don't have the time (as in true snapshots), "P" mode gives usually the best chance of getting technically ok photos.
When walking around, my camera is normally set to "P" mode for above reasons. Shifting the program to achieve a specific time/aperture is, at least with the ergonomics of my camera, normally easier than changing exposure "modes" on the fly.