Ugh. I wouldn't recommend animating in photoshop; it's much more difficult to control the exposure of the frames. (i.e., it's difficult to animate on two's, which is standard for 2d animation) However, photoshop will export to most major video codecs.
I don't know anything about flash, but both AE and Premiere have layer blending so it shouldn't be a problem to set a top layer to 'multiply' over a bg.
I would recommend compiling your animation in a program like Premiere or a free program like MonkeyJam (I personally have flipbook, which is pretty simple to use but not free) Once you have the video exported you can then drop it into AE.
If you try and compile the drawings in AE for animation, you'll end up with a layer for each image - which can make it difficult to navigate/edit when you're dealing with more than a few seconds; vs. Premiere where everything is on one layer (as it's a non-linear editing program) or monkeyjam/flipbook which are designed specifically for animation so they mimic the classic animation exposure sheet.
I can't comment on Flash, since I don't use it. I'm not a fan of the drawing tools personally, but studios like Titmouse have used it for frame by frame animation. Obviously if you're animating in flash, there's no reason to use a program like monkeyjam or flipbook, though.
Good luck!
Oh. by the way. If you are animating traditionally, camera moves need to be animated on one's - otherwise they can appear jerky.
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