The quickest way that I've found to take up vinyl tiles is to 'shave' them up with a garden spade or a specialised tool from a hire centre, finishing the last awkward bits with a hot-air-gun and a strong (not thin and cheap) scraper. Ventilate with the heat. You could also try mineral turps to soften the old adhesive. On a wooden floor you have to be careful, of course. Small indentations in floors can be filled with builders bog, similar to car bog, but that is an expensive twin-pot, so there is mixing involved. Large filling is a cement floor-leveller job, which is only available (here) in 20kg bags, so expensive and wasteful for small jobs.
If the concrete floor is in direct contact with the ground, I'd redo the painted seal. Talk to your paint specialist about that. Any irregularities will inevitably show eventually as you walk on it.