I've been told by several vendors that sell sharpening supplies that the only thing that will cut carbide is diamond. If you are using high speed steel gravers then almost anything will work. That being said I would say that in my very limited experience that sharpening is the most important thing you have to get right. I struggled to get anything to work until I got the graver geometry down and a nice, sharp, polished graver. With that you can make nice clean cuts by hand, with a hammer and chisel or with a power graver. With a dull tool it's not going to work. My advice is save money on something else and buy quality sharpening implements. There are numerous systems out there and searching this site and other engraving boards should show you where to go.
I would say you can but the step from the coarse 800 grit to 3000 grit in one step is more than is normally done. It will take a while to get the scratches out with the 3000 grit. You may want to polish afterwards with some rubbing compound on a board or leather strop. Do your research on this forum on graver geometry and fixtures. Sharpening successfully freehand is difficult, especially when first starting out. If you can't afford the fixtures you may be able cobble something together to get started.