I think a drop sheet would be a waste of time if you have tiles. You will have a big mess either way - you will have to clean the tiles or the drop sheet every meal time, so you might as well just clean the tiles.....

A spray bottle with vinegar/water in it for wiping down after is good, as are lots of cloth bibs/face washers. As long as you are diligent about cleaning up immediately, it won't really matter what sort of highchair you get - they all get filthy. However if you are a bit slack like me, a solid plastic or wood one might be better if you sometimes "forget" to clean up right away. Those cushioned ones are much harder to clean with all the crevises etc.