How to repair chipped tile edges on an external facing corner

Scraper. This is going to be a tough job if you want to keep damage to the wood to a minimum. It does look like the floor has some type of finish on it before the grout material was applied. This should help a lot to make the scraping removal a bit more practical.

Note that what ever type of scraper that you use will likely need replacement or resharpening a number of times in your removal process.

I honestly don't even see how a poorly done tile job can leak this quickly. I can tile and grout a wall pretty sloppy and it will be good for at least 4-6 month and possibly a lot lot longer.

The short-term fixes you can do now are exactly that at best - short-term. It is possible that you will spend money trying to waterproof after - which in no way is ever the answer - and it could possibly have no impact.

You need to get your money back from this contractor. Even a bad contractor puts up some sort of water barrier in a shower. A bad contractor would put up the tile with a bad grout job or wrong grout and maybe the drywall gets wet and eventually moldy. Your contractor either has no clue or is insanely lazy. You cannot let him back in your house.