I'm not familiar with the integral cup, but it sounds like it would be a real nuisance to use, as not all spraying is done horizontally. If you're using water-based paints, you could use an eyedropper or a plastic syringe (minus the needle) to transfer paint to the cup, or, when you're finished painting, back to the jar.
For thinning, I mix the paint in small jars. My airbrush, a Paasche VL, came with one small spray bottle, but I found that the cap is the same size and thread type as the bottles for Floquil, PollyScale, Accupaint, and Testors. As an added bonus, the old small Polly S bottles are exactly the same as the Paasche bottle.
Most of my airbrushing is done with Floquil and I usually try to mix enough paint to do the job, although the shelf life of the thinned paint seems to be pretty good. Right now, there are 19 small (5/8 oz.) and 2 large (1 oz.) bottles of thinned paint on the shelf in my spray shop. Some have been there for a year or so, while others have been around much longer, as common colours are continually being added to or altered.
For thinning, I mix "by eye": paint first, then add thinner. For solvent-based paints, like Floquil, I use lacquer thinner, poured from a 1 litre can, which can be a bit messy, but it's a lot neater than pouring from the gallon can that I buy it in.

If you're using water-based paints, use the cc graduations on the syringe to keep the paint/thinner proportions constant.
Another bonus with using pre-thinned paints stored in bottles is that there's almost no need to clean the airbrush between colours. Exceptions to this would be light colours or clear finishes, as even a drop of a darker colour can alter these.
Finally, when you've finished painting for the day, CLEAN YOUR AIRBRUSH THOROUGHLY!
Don't leave it 'til later, it'll just get harder to clean. My VL can be torn down, cleaned, and re-assembled in about 60 seconds.
Wayne