The 135mm focal length is fairly long on a crop sensor camera and doesn't have IS, so you need to watch your shutter speeds... keep to about 1/200 or better unless you are really, really steady holding your camera or using it on a tripod.
it's a bit more hand-holdable on a full frame camera, though you still have to be concious of it.
FF camreas simply have smoother tonal gradations than the croppers. And this particular lens is all about soft backgrounds, bokeh and tonal gradations. So, while it works wonderfully on a cropper, IMO the 135mm really comes into it's own on a FF camera.
No, I wouldn't call it the sharpest lens Canon offers by any means. It's a dreamy, "artistic" lens. If you want really sharp and an almost "clinical" look, get a 100/2.8 or 100L macro instead.
I think it's the difference between the crisply focused areas and the heavily blurred out of focus areas, that makes it seem "sharper".
The 200/2 is sharper than the earlier 200/1.8, but it's not a huge difference. The main difference is the addition of IS... and parts supplies are exhausted so you probably can't get the AF fixed on the 200/1.8, if it breaks. The 300/2.8 IS is very, very close to 200/2 sharpness. The 200/2 on a FF camera can blur down the background more than the 135mm can on either FF or cropper. But both are fantastic in this respect.
135/2 on 7D:

After the show
EF 135mm f2 lens at f4. EOS 7D at ISO 200, 1/500 shutter speed. Handheld, available light. And on 5D Mark II:
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Lost in thought
EF 135mm f2 lens at f2.0. EOS 5D Mark II at ISO 6400, 1/200 shutter speed. Handheld, available light.