Polishing you mean sanding or just varnish? (Solid Wood)
If you need sanding (Meaning have uneveness, cracks, chips) you will need to sand one layer first then do painting.
If only varnishing, you should do painting first becoz varnishing is the final touch.
1) i have a painter (Old Uncle) who is quite neat and tidy in his painting works, especially for stay-in painting works. Suggest you get some quotation for comparison first then ask around for pricing to have better idea of the market rate, anyway you still have some time before the dates to start. Perhaps you can get the Nippon painters or ICI painters for a start? They can also give you useful advices. Its no harm coz ultimately you will be buying their paints as well.
2) Homogenous tiles cannot be polished. If its scratched its scratched. Only natural stones like marble and granite can be polished.
3) So if you need sanding of solid wood, do that first becoz its dirty and sandy. After that paint whole house, then final varnishing to the shiny or semi gloss finish.
Btw i have contact for the wooden flooring guy also. If you need please send me a PM.