yes you can do it with posts between teh existing posts. then as you have time simply remove/replace 1 post at a time until the rotten ones are gone.

as to preventing rot, use PT posts and make sure you put the post in the ground first, then a layer of gravel, then pour the concrete. this way water can drain away from the post. remember that wood is pourous and the water that hits it up top will week all teh way to the bottom of the post. if it is 100% encapsulated in concrete, it starts to rot.

Similarly, I put about an inch of gravel at the bottom so the PT post isn't sitting in water.

I want to build a 8ft to 10 ft high fence. How deep and wide should the hole be, and how much cement should be used?

How deep and wide depends on where you live. If you have wet/freezing temps consistenet in the winter, you want to go below the frostline in your area. You can get that info from your building dept.

If freezing isn't a problem in your area, you have to consider the soils type. If it's loose, then you are going to want to go wider but not necessarily deep. Making the hole wider at the bottom and narrower at the top helps to keep from rocking. Too loose of soil you can go wide and fill with compactible filler soil and tamp the bottom and sides.

In real rocky soil, non freezing, you can typically get away with minimum width and depth but in all cases, dig it out deep enough and wide enough to get some gravel in the bottom as well as around the sides after setting the post before concrete for drainage as mentioned above.