That does not look like a load bearing wall to me. I would expect to see studs in the middle, or a header at the top. Here is a drawing of a small bathroom renovation I am doing right now. The room is only 5'x6' but I am squeezing a shower in there too. I bought a wall mount toilet and wall mount sink to save space and keep the floor open. Combined with large format tiles I am hoping it will make the room look bigger. Just to give you some food for thought. Definitely get rid of that wall to open things up. Also consider a smaller vanity so it doesn't take up so much of the room. Maybe a bigger mirror behind it and some wall shelves to make up the storage.

No trench drain at the rear? No capillary break? No wetroom waterproofing of the whole chamber? https://www.ceramictilefoundation.org/homeowners-guide-to-hiring-qualified-tile-installer

As has been mentioned, the most common cause of water hammer is the quick open/close action from fast-acting valves. Typically found in washing machines and dishwashers. It's less common with toilet fill valves, faucet taps, or shower valves. First thing you can do it to drain your entire house supply plumbing system. Normally water hammer arrestors are plumbed in above the valves during construction. You may have some already in your house's plumbing system. Over time they can fill with water. Draining the plumbing system can drain the water out of the arrestors, and when the supply is refilled, the air pocket that provides the cushion is reestablished. Turn off the water supply to the house. Open every faucet of valve in the house. Then go back in the basement and open the drain valve, or needle bleed valve(s), or basement utility sink tap...you're just trying to drain the hot and cold copper supply tubing above. No need to drain the water heater, boiler, etc. Just your domestic water supply tubing system. When drained, close the drain valve(s), turn the water back on, then go back around closing all the taps/valves. It took about 20 years before I had to drain my system to recharge the arrestors in my house. Last year I was in the shower and the water started surging a bit and I heard a series of mild "thump" noises. That had never happened before. My wife had started laundry. The thumps coincided with the washing machine valves cycling. I drained and refilled, the issue disappeared, hopefully for another 20 years. FWIW, when I plumbed my house I used simple extensions of copper tubing for water arrestors, similar to what you linked to in your photo. Also check your household water pressure. There should be a pressure regulator on your well's water supply tank, or at the point where city water enters your house. House pressure is often regulated between 40-60PSI, or 50-70PSI by well pressure regulators. City water can be higher, you can regulate it down to around 50-60PSI. If neither of those work, then attack the change in your system; the new shower valve. Since it's already tiled in on the shower side, I'd open up the back of the wall if able. Review the plumbing layout and have water hammer arrestors installed on both the hot and cold sides above the valve. If that can't be done, then the Watts LF-150A that Geoffrey linked to is a good retrofit option. You'll want one each on the hot and cold sides. They can be installed under a tub, in the basement, under a vanity, etc. I've never enclosed one of them in a wall. They may not cure the issue, but they can lessen it. Agree with Mule Meat, the "unused parts" in the small ziplock look like port plugs for your valve. If your installation did not require all the ports on the valve to be used, you simply plug the unneeded ports with those plugs. The green colored sleeves look like wall anchors.

I would do a cabinet with the sink rather than over the toilet. I have three pics to show you. And could have or could show you billions. The first two are layout ideas. Please do not focus on the style of he pictures. As that is not what I’m showing. The third is a small bathroom style idea which I love. And think it opens up the space as much as it can. Uses the potential. And you can do the same. Tile depends. I would probably say to stay super neutral and do all the walls and more intricate floor tile.