If your camera can both take long exposures for imaging and short exposures for guiding then it is possible. To my knowledge no such camera exists though. You should be able to get good images by stacking LOTS (I mean hundreds) of 2, 3 or 4 sec exposures such that those exposures are used for guiding at the same time but then you'd probably lose a lot of SNR because most cameras work best at longer exposures. So in general I'd say that it is not possible or at least strongly not advisable.
It is possible if you have a camera with non-destructive readout.
We don't have that.
Think of it like this: If you have short enough exposure times that you - can- guide with your imaging camera, you -don't have to-, because the tracking error would be negligible in that case.
Best is to simply get a guide scope or off-axis guiding. Personally, I prefer a separate, light weight guide scope, because it's more light sensitive and it prevents issues with vignetting on the imaging camera. Also I already have a lot of weight on the focuser.
Agree, and the only reason to have an OAG is to take advantage of the higher power of the imaging scope and to avoid flexure. If you can control flexure, the first problem is not so much a problem anyway, since seeing changes are usually larger than the resolution of a 280 mm guide scope. I.e. if I get an RMS of 0.7, and seeing limits resolution to 2.5 arcsec/pixel, any 10% gain or so by using an OAG will be negligible.
There are very high end cameras that contain a guide sensor above the main image sensor. But that's really just a glorified Off Axis Guider.
So the answer unfortunately is No.
One attempt to do this was a camera with an interline sensor which would read the camera data in two separate passes, one for the odd lines and one for the even lines. I don't think it really caught on because of the obvious problem that the image resolution was halved in the line direction. Amplifier glow would have been another problem, you can't turn that off if you are still reading the camera.