Sure you can, we do it all the time in rental properties. Old houses have windows in the showers and renters do not take care of tile. You just need the right saw and then tub trim. Not everyone can tile- it takes a lot of patience and a tile saw. And if you are not a DIYer naturally it can be a big mess. We have done a lot of repairs to walls from botched tile jobs. And hiring out is expensive. You say you have a small budget. Chances are with the look of the tile in the pic- you may have water damage already. Powdy, it all depends on your skill level and actual budget and what you find underneath you remove the tiles.

Well I am a very plain sort of a person and I don't like overdoing patterns. The timber in the cupboards is beautiful and I feel that bench top would detract from it. I would have as plain as possible top in a cream/beige to enhance the timber. This would give you plenty of choice of other colours in the room. I don't feel I can really comment on colours as I can't tell the aspect of your room or the lighting nor do I know what other colours you have through the house. The only downside to the cupboard doors you are considering is that they will attract dust and being in the kitchen a greasy dust.

Maybe if you stuck to one colour scheme such as white and purple or grey and lime green, colours that compliment each other. You could have a cabinet on the larger side with some family photos or a mirror above it and maybe a single colour on the top half of the wall and a complimentary wallpaper on the bottom half. You could go for wood on the floor or a simple soft carpet but it might get muddy. You could also have a bench with storage underneath for shoes and bags, but try get a colour that matched the walls. You could also paint the banisters if they don't match the rest of the room.

If I understand correctly you are adding extra floor by bay window feature ? Have you thought as out a window seat to hide the floor issue, can give you more storage and a really useful seating feature (great with kids)