It's easy for me to say I'd go for it, but I could fix it if it broke, and I kind of have the muscle memory to know "this is gonna break if I have to wrench on it harder". If one of my employees just added a tee, I'd be irritated because it would be lazy work, and the customer doesn't hire us to do lazy work.

But! Adding another tee is 100% possible, and while I would never do it, you definitely can and I give you my blessing to just add the tee and leave it at that 👍 it'll still be better than government work!

Please do!

Hard to tell if it's brass or not, but I'm leaning towards yes. The angle stop definitely is. If you rub sandpaper on the pipe it should look gold if it's brass, silver if steel. If you have a nice strong magnet to test with the brass will not stick to it. That link I posted is for a PVC transition but it says it's also for ips. PVC, iron, steel, and brass piping all have the same outside diameter, so it should fit. I've never used an IPS one because I've never had to and in general I prefer not to use shark bites (they very rarely fail but are not the most reliable method of repair) but from the description it does say it works for IPS. If you do have to use a shark bite, the pipe must be damn near perfectly smooth and without scrapes or burrs or else it will damage the shark bite's o-ring and cause a leak.

I don't think you'll have an issue unthreading the old valve, I just want to give you some emergency options.

In 13 years I've never had threads "come off" with the pipe. The only thing I've had happen is the pipe would crush when I beared down while removing. Those pipes I knew were unlikely to hold up though because they were so corroded, and were steel-steel. Yours looks like it's in remarkably good shape to me and I would have no worries about it. Brass-steel is much easier to remove and brass-brass even more so. That said, nothing in life is guaranteed and yea you might be in trouble if something were to break. If it comes to that, you can cut off the damaged threads and after sanding/cleaning any scratches or scapes off the pipe use an IPS (iron pipe size, same outside diameter as galvanized steel, brass, and PVC) shark bite like this and continue your work exactly like you would have done, or if it's a hard to find item you can probably more easily find a shark bite 1/2" IPS cap to install until you can get the parts you need. Rethreading the pipe is also possible but not an option I'd go for at that point.

Your best bet to avoid damage is to make sure your wrenches are secure while you're removing the old shut off valve, and put your leverage into the wrenches, not the pipe. I see guys put their weight into the wrench and it puts stress on the pipe.

Short answer, no, not without extensive work. You're better off hiring a licensed plumber for that kind of addition if you want to do it right.

Another thing to mention, when you remove the old valves from the pipes coming from the wall, you should use two wrenches. Ideally, a small pipe wrench for the pipe coming from the wall, and an adjustable wrench or tongue and groove pliers for the valve. Two t&g pliers would also be just fine if you don't mind some cosmetic scratches.

One will be used to remove the valve, the other pair will be placed in the opposite direction to HOLD THAT WALL PIPE STILL

Don't let the pipe coming from the wall rotate or you might have to unthread it completely and reapply tape/dope to reseal it. Speaking of tape and dope, tape should only ever be applied clockwise, otherwise tightening a new fitting can unwrap the tape. So when you remove that valve, looking down the barrel of that wall pipe, you'll apply 3-4 full wraps of tape in a clockwise direction. You won't need pipe dope, but I always smear a real small amount onto the tape. It makes it take that much less effort to tighten things together. A sharp knife is less dangerous than a dull one, and you'll need less force in an already awkward position (working under sinks is fun!) to properly tighten things up. Think snug, not cranked on. There's nothing to be gained from overtightening.

Plumber here, here's how I would go about it.

Those are either 1/2" brass or painted 1/2" galvanized steel pipes coming from the wall. They should not be black iron, as it was never ok for water, only gas. If you're curious if they are steel or brass, a magnet won't stick to brass. Either way, they both have a 1/2" threaded connection.

The line on the left is hot water, the line on the right is cold. Your dishwasher should be tied into the hot water line.

I would start from scratch and replace both stops.

You'll want 1/2" FIP X 3/8 compression angle stops. You posted a 1/2" compression stop elsewhere, that won't work. You need the 1/2" npt female threaded type, which will probably be listed as 1/2" FIP X whatever size you want to use. Let's go 1/2" FIP X 3/8 compression for both hot and cold. (Note the shortcut below). These angle stop's threaded 1/2" connection to the wall pipes will be the only place you use Teflon tape. Compression and rubber seals do not require it and you might compromise their seals if you do.

Hot side: after installing the angle stop, get a 3/8 X 3/8 brass add-a-tee. One outlet will be for your dishwasher, the other your faucet. Your faucet hose might not reach now, so maybe get a 12" (or whatever length reaches) 3/8 female comp X 3/8 male comp stainless steel braided supply line to bridge the gap.

Cold side: the shortcut would be if you can find this guy. The 1/4" outlet will go to your fridge. The 3/8" outlet, you will add a 3/8X1/4 add-a-tee. The 1/4" port on the add a tee will attach to your white plastic water line (there should be a stainless steel or brass tubular insert inside of the plastic pipe to avoid crushing it when tightening). The other 3/8 port will go to your faucet. Again, you may need another 3/8 female X 3/8 male comp supply line to extend it to reach.

If you can't find the shortcut double angle stop, get another 3/8 X 1/4" add a tee.

That should do it! Hope this helps, I love you.

You're worried that the overhang of the washer wobbles too much? You can put another clamp onto the rod, and have it meet the washer to back it up. Being that close to the edge of the wood you'll want to pre-drill the holes for the wood screws to prevent splitting.

Yea, agreed it's maybe a blockage. The line would have to have completely separated to have lost pressure. You wouldn't notice a pressure drop from anything other than a massive leak.