I can't say if one can buy the one's that I have in a single (I don't have any single beds), however, those that I do have I got from John Lewis. As far as I recall (been a long time since I bought 'em), they were around £70. They may be a bit dearer now. With these, sorry I can't tell you the name of them, I threw the boxes out long ago, one can have either one side, or both sides on. Both sides function separately.
They have several functions: i.e. you can set them to be on all night, or on a timer, or just manual switch. Every here has stated correctly; with the John Lewis one, and probably all other brands, you can switch it on, go brush your teeth, and find it has warmed the base nicely (yes, they are to go on the mattress). No, they can never be used folded.
I don't believe that I have ever heard of any that go on top of one, under the duvet; doesn't mean to say that one can't buy them tho, just that I don't know of any. They have drawstrings that go under the mattress, sides, top and bottom, and tie up at the side of the bed. Personally, I use the manual turn off/on action for mine, simply because I'm technologically challenged by these things; Heck, I can't even figure out the DVD settings... I always turn mine off when getting into bed, I never sleep with it turned on, not on any setting; altho the instructions and settings do have an all night function.
Everyone is different in their requirements regarding duvet togs. The one I have on my bed is a 5.5 tog, and I find that even too hot during the night. I like to have a window open in the bedroom all winter, day and night. During the course of any one year I tend to have a lot of guests, and as I said everyone has different requirements; so I have duvets ranging up to 10 togs. But I could never sleep with any above the 5.5 tog. I only wish that one could find a 4 tog, but it seems that they are not made; at least I haven't found one yet; tho I haven't given up hope.
I'm sure they must do a single bed size in the John Lewis one, and I can highly recommend them.