Firstly use flooring grade ply 4mm thick. Then if it’s a cushion vinyl you don’t need to stick it necessarily, but standard sheets vinyls or vinyl tiles must be fully adhered.
This is not such a bad idea, but as a flooring contractor I would investigate the use of fibre board instead. It will give better thermal and acoustic properties and probably won't cost anymore.
Your sander is correct. Just mask up the floor when painting the skirting boards. This will be much neater and quicker, and definitely the way our flooring company in Horsham would do it.