Electronic – Should I solder multiple wires together, or is there a better way

Flux. You're missing flux, and lots of experience. I regularly solder with a tip that easily covers 10 or more pins at a time, but by carefully controlling how much solder is on the tip and using lots of flux I can get perfect solder joints. When soldering a TQFP or TSSOP package here's what I do:

1. If the pads on the PCB are not super flat, then use some solder braid and suck up some of the solder on the pads to make them flat.
2. Put the part on the PCB and carefully align the part. Don't rush this step, because if you mess this one up then nothing else will work correctly.
3. Put a drop or two of liquid flux on the part. I use water soluble liquid flux, but others are successful with other liquid fluxes.
4. Clean the iron tip and put a little bit of solder on it.
5. Carefully touch a corner pad (NOT the pin itself) and let the solder wick up and tack down that pin.
6. Repeat 4 & 5 for 2 to 4 corner pins. Reapply flux as required.
7. Put some solder on the tip and run the tip down one complete side of the chip. Ideally you won't have bridges, but if the last 2 or 3 pins are bridged then you did good. Ignore the bridges.
8. Add more flux (you can never have too much flux).
9. Repeat 7 and 8 for all sides of the chip.
10. Add flux, clean the tip on the sponge, and touch the tip to the bridge. Solder should stick to the tip. Repeat until bridges are gone.
11. In bad cases, careful use of a solder wick can help. But be careful because it is easy to bend the pins and then you're, um, hosed.
12. Clean the flux off. Some flux can be conductive, like the water soluble flux, so make sure it's clean (and dry) before powering up.

I should mention that my soldering iron has a straight, but angled tip. It's often called a chisel tip. This tip makes it easy to do everything from 1206's, 0402's, TQFP's, TSSOP's, etc. It just doesn't work for BGA's, QFN's, and some of the weirder things.

The first thing I would try is a spring, like the ones used in those old Radio Shack electronics kits from the 80's.

If you can source one long enough, then it might be pretty easy to remove a wire from the middle, provided that you have reasonable spacing between layers.

Maybe some continuous-length extension springs from McMaster will do the trick.