If you are using C track and yet to buy a turntable, I strongly recommend you buy the C track turntable.

To answer your question, yes you need to use the adaptor track(s). Might be better to do your whole BW area in K track.

No you cannot use the C track extensions on the K track table.

The way i would try to do it is to use rail joiners between the radial pieces on the turntable and the C track pieces. There is a subtle problem though, K track as used on the turntable uses code 100 track, and C track is code 90. So there will be a 0.01 inch difference in height between the top of the rails if the bottom of the rails are even.

Some trickery with the rail joiners can sort this out (Peco make joiners for connecting code 100 to code 83, which is what I would be tempted to try, but that is almost as much the other way) or one could just put up with the slight clunk across the join, or try filing the rails.