I'm considering purchasing a house where I'd want to remodel the bathroom. I know there are plenty of plastic and freestanding options for tubs but I can't find any resources for using tile and grout to create a tub.
You can make a tub out of tile.
There are some issues to think about:
you will need a thicker more rigid structure for the tub and they can be made custom to an area and look great.
The massive amount of tile and concrete cools the water very quickly which can be mitigated somewhat.
I tried several methods and found a water jacket water heater with a recirculating pump was the best for keeping the tub from cooling off too fast just like a garden tub or hot tub uses. I had also tried using a space heater to circulate warm air around the tub but the base did not get fully warm. I used a separate temp controller in both cases. The air temp had to be at 100f and that made the bathroom uncomfortably warm and bottom of the tub was cool for a while. If I were to do it again I would make the tub base and walls even thicker and embed a hydronic heating system using a separate 5-7 gallon water heater and recirc pump only running the hydronic system shortly prior to filling the tub.
My speculation is the reason you can't find resources for that specific approach is because that approach is inferior in almost all ways to modern tub construction.
Most importantly: tile and grout is mostly water-tight; fiberglass or cast-iron tubs are entirely water-tight. As you probably know, any amount of regular water leakage into unintended areas can cause huge problems. This is why tile and grout is fine for areas with indirect water (e.g., tub surrounds) - though this still should be backed by a water-barrier (e.g., Redgard) and concrete board. Even when you see tile and grout shower floors, these are always underpinned by a "shower pan" - which is 100% water-tight. Tile and grout is, to some, a better aesthetic choice, but it is worse in the practical considerations of water control.
There are other, less critical reasons that it is avoided: one, it's generally more expensive to manually tile and grout anything than it is to purchase a prefabbed, mass-produced version; two, the curves and smooth presentation of almost all tubs is much more difficult to accomplish with tile and grout; three, a smooth fiberglass wall is much more comfortable to move your bare skin across than tile and grout (ever try and slide across your kitchen floor on your stomach?). All of these can be mitigated with highly skilled craftsmanship, but that's much harder to find than a tub at your nearest hardware store.
You do see tile and grout approaches in some large-scale water containers like public pools or spas, but even these are almost always built on top of concrete and dirt (where the small water leakage is controlled or irrelevant). Here the tile is for aesthetics and durability (a porcelain tile will wear down much slower than a concrete wall due to its hardness).
To summarize and directly answer your question: it can be done, but it will be far more difficult of an approach than any standard method because the installation will be customized and extra considerations will have to be taken to control the water. At the very least, it should use porcelain tiles (more water-resistant than standard ceramics) on top of Redgard/Redgard-equivalent on top of some kind of water resistant substrate (concrete forms, concrete board, shower pans, etc.). The last part will be the most difficult and particular part of the process, as standard construction materials will not adhere to a typical bathtub shape.
I have no idea. Affordability is relative to the size of your bank account.
Probably not, they don't seem very comfortable. Do a Google search for "tile tubs" or just see the image below. If you get inspired by the image then make sure to give the outer edge a gentle slope into the tub.
Water-proofing is probably on-par with a fully tiled shower.
I hope you use a dark grout because the base of the tub will be noticeably grungier than the side walls in very short order.
I hope you enjoy cleaning and sealing grout because that will consume a few hours of your life every month.
Watch your step! If you slip and fall then I think the chances of a life-altering injury are markedly higher.
The real question you have to ask yourself is whether you are helping or hurting the value of your home. I would say that a tiled tub will be utterly shunned by parents who have or plan to have children. Ditto for anyone that might be interested in getting a walk-in tub for their later years since they'll have to demolish everything and start from scratch.
Whenever you do something to your home you have to ask yourself whether other people will see it as a feature or a "what's that?"
If you really want to do that (and only you can judge), I'd recommend to use stoneware instead of tiles. Stoneware is coloured through and lasts longer than tiles, which have a thin coloured layer on top of a white body which gets rubbed away with use. Also, stoneware is available with rough surfaces that are less slippery when wet.
You'd also need a water-proof layer between the concrete structure and the tiling cement, there are latex-based paints for that. Those need to be applied very carefully especially in the corners, or water will seep into the concrete, leading to moulds and structural damage. Definitely a job for experts.
And do make sure you have sloped walls and floor (!), so the water flows completely towards the sinkhole when you empty the thing.