Is it okay that my soldering iron just sits loose on the desk between joints?

You're not overthinking it, the rag is the part I'd fix first. It's worked so far beacuse the iron cools down between joints, but a hot tip parked on fabric for even a few minutes while you hunt for a part can smolder thru it. I did exactly that years ago with a folded shop towel, smelled something burning, and found a neat little charred ring underneath. The scary part wasn't the towel, it was that I had a plastic parts tray sitting on the same towel. Fabric also holds heat and it slides around, so the iron can roll off the edge of it onto whatever's next to it, and you mentioned that's a tangle of wires and a board you care about. A proper coiled stand is the fix and it solves your space problem too, not the opposite. The spring type is basically a dome maybe five or six inches across, sits anywhere, and the iron drops in at an angle so the tip points up and away from everything. Most come with a little tray for a brass or steel wool tip cleaner, which is nicer than a damp sponge on a tight desk anyway. Cheap enugh that there's no real reason to skip it. You don't need a fancy station, just something with actual weight where the iron can't wander.

On clearance, a few inches each side is fine for the kind of work you're doing. The thing I'd actually move is anything melty or pressurized, so if that hot glue gun has sticks sitting right next to the iron, shift those a hand's width further out. Wires are less of a worry than the insulation on them. One other trick for tight spaces is to give the cord a loop anchored to the desk edge or taped down, so if it gets tugged the iron pulls toward the stand instead of off it. So yeah, buy the stand, keep the rag as a wipe cloth for the bench, and stop tensing up mid joint. Tense hands make cold solder joints.

Back in my apprentice days nobody bought stands, we made them. Tin can with a wet sponge in the bottom, or a bit of coat hanger wire bent into a cradle screwed to a scrap of 2x4. Heavy base is the whole trick, that's why your sheet metal one walks off the desk. The bent wire trick still works fine if you want to save your money for a decent iron. One thing I will say after forty years of soldering, get the habit now of never laying the iron down anywhere except the stand, not even for a second. Old habit, saved my bench more than once.

I've been using one of those flat square tiles from the hardware store under my iron, the kind you'd stick on a wall. It doesn't slide and the hot part never touches anything flammable, but the iron just lies there pointing across the desk so I get why everyone keeps saying get the spring thing. Does the iron care wich way it points in the stand, or can the tip face sideways if that's the only direction I've got room for?

For what it's worth, where you put the stand matters as much as the stand itself. Mine sits just off to my left and slightly behind whatever board I'm working on, so I can drop the iron back in without even looking. Took about a week to build that habit and now I don't think about it at all. If yours ends up on the crowded side next to the glue gun, you'll be reaching across stuff every time, so maybe shuffle the desk around before you buy anything. Also a thick rubber desk mat under the whole area helps a ton in a tight space. Catches stray solder blobs and stops everything sliding around.

Following this with interest since I do my soldering at the kitchen table and pack everything back into a shoebox after, so my setup is even worse than yours. The cord loop trick from above is something I never thought of, might try that. Question for the OP, do you leave the iron plugged in the whole session or unplug between joints? I've been unplugging whenever I step away but I don't know if that's overkill or acutally the bare minimum.