How to prepare walls for wallpaper after removing wallpaper

You will need to take off all the glue if you want a good finish.

The paint will lift the glue, but unevenly, and mix with the paint. This will cause the paint to clump and not cover properly and all in all it will look terrible.

I know because I did this in the first house I owned and had to wash the walls of the one room where I tried this several times to get as much glue off as possible. I don't think I got it completely off and you could still tell.

If you just want to paint the room, you would be better off hanging lining paper and then painting that. This has the added advantage (especially if you use 1000gsm paper) of covering any imperfections in the wall.

I also think removing as much old adhesive as possible is what you want to do. I made the mistake of not removing it all a few years ago and now I have to go back. Also, seal with kitz oil based primer. The skim coat sounds like a lot of work to get it smooth. Is the wall that rough? Af for the need to prime, definitely after skimming, or on any holes you patch. I wouldn't prime before skimming unless there was still glue left.