1. If you have a .png, save it as a .bmp or .pcx another compatible format that JCS can use. I don't recommend using .jpg for tilesets, as it's a lossy format and doesn't play nice with pixel art or limited palettes.
2. A tileset will always be 320 pixels wide, or 10 tiles, as a tile is 32x32 pixels. The height needs to be a multiple of 32, and you're limited to 32,640 pixels in vanilla 1.23, and 130,880 pixels in TSF. It's unlikely you'll need a set that large, though.
3. Essentially any program that gives you control over the image's palette in some way can be used. I've seen Paint.NET suggested before, and I personally use GraphicsGale. First priority is getting the essential, tileset-specific functionality, and from there it's more a matter of personal preference.
4. GraphicsGale does indeed have a grid function, which you can specify the size of.
5. A tileset is limited to a 256-colour palette, and if you have more (and so your image isn't 8-bit) then JCS won't create your tileset file. Jazz 2 itself uses several entries from the palette (I think Disguise's Tileset Resource probably covers these?), and while you aren't forced to keep them the same as usual, it's generally better not to change them as it'll affect all the sprites on levels with your tileset.