The leaks didn't come from the shower head/ faucet side but on the opposite end, which is an outside wall. The water was dripping directly underneath down the basement wall. I've scoped out a couple of pro tile installation/repair companies and will have them come in to check it out and provide estimates to clean the floor and repair the shower. Finally got up to the attic to get the spare tile. Fortunately, I've got about 70 brand new wall tiles which the original owner had left behind. Gathering dust for over half a century but am glad to have them!

Cement board will work fine for a hearth. However, you will probably need two layers since most codes require at least an inch of non-combusitble material on the hearth (check your local code). It's highly unlikely that the hearth has any structural connection to the fire box. However, there could be steel that is connected. You may want to use a concrete saw to make a clean cut at the junction of the hearth and firebox. As an alternative, you might want to consider putting down cement board over your existing concrete hearth and tiling it. That way you save a lot of demolition, the hearth will only be raised slightly (1/2"?) and the tiles shouldn't come lose since they will be on a solid sheet of cement board.

That's very cool.

It looks like those gas logs are open to the room. this means you will never get rid of the soot. I would replace them with a gas fireplace insert, its much safer since its sealed, cheaper to operate, will warm your home, and not put more soot on your stone. You can pick a front that compliments your home. i believe your stone is sandstone, and can be cleaned successfully, but i do not have that knowledge. suggest joining some mid century enthusiasts groups on FB, lots of expertise there if you dont get a confirmation here. Best of luck.