Long Zoom for A6000

The best APS-C telezoom in the Sony line-up is the 70-350. But your looking for a lens that also covers wider focal lengths. If you need 200mm, the 18–200 mm F3.5-6.3 OSS LE (SEL18200LE) fits your requirements best. AFAIK all the 18-200 lenses perform about the same. You can also take a look at the Tamron version of the LE.

Having said this....

Get the 18-135 OSS. It is a really good lens and you can crop the last bit if need be. Really, skip the 18-200's, get the 18-135.

Did you look at the lensdatabase? It will give you some more experiences.

The cheap-end zooms vary a lot. In A mount the 18-250 is a lot better at the long end than the 18-200 (which is an old Tamron lens) but the 55-200 is a lot faster mid-range than the 18-200 (about 1 stop faster), and most of the 18-xxx lenses tend to shoot straight off to f6 or so as soon as you leave wide angle end.

Plus the ultra-zooms generate a lot of distortion. If you take jpegs, buy a Sony and get the built-in corrections (though I think some of the non-Sony E-mount lenses do offer corrections too?). If you take raw and do your own corrections, not an issue though you need to allow for the loss of edges at the wide end. So if you don't need the wide end, don't buy it!

I have the 55-200 in A mount and E-mount and both are good. The E-mount is quite a different lens design, of course, and the OSS is brilliant. I would definitely recommend that one.
I've also tried quite a few mirror lenses in A and E mount including the Kenko 400mm mirror which is compact and very sharp (for a small mirror...), but you won't get 2x more definition from it compared with the 55-200mm and 2x crop/dig zoom, because the latter is very sharp and mirror lenses always lose a bit in the middle of the MTF curve.