Mot: I have been having good results with Kester 63/37 solder (mine is labeled organic core - whatever that is). A friend has noted that the Wonder solder is very easy to work with (quick joints with a 35 watt iron).
The best thing to do (regardless of how skilled a solderer you are) it to use plenty of heat sinks when working around caps, voice coils, et cetera. I currenty use the little copper plated clips (with flat ends) from Radio Shack. I used to have "solid" copper clips (which were even better), but misplaced mine and cannot find a new source for them (these were @ least 50 years old:-).
In general I prefer solid core/Teflon wire (over stranded wire) in the signal path. Stranded wire can produce a better defined musical image, but the downside is that it seems to smear transients. I am happy with the slightly blurred imaging but super clean/fast transients of solid core cable in my system. I have also found that in order to get a good balanced sound out of solid core the guage should be no larger than 24 (I use 26 guage myself on everything except the phono cables, which are smaller).
I have experimented with over sized (way larger than the wire) teflon tubing (running bare wire through it) and this sounds great for IC's, but unfortunately does not work well in speakers due to its tendency to vibrate, I think. What you will need is very thick Teflon tubing for this application (which sounds better in speakers than the very thin tubing (the type that you can peel off with a finger nail) and this again may be due to vibration/physical damping I would guess. If you want a more open and 3D type of sound avoid twisting and/or braiding the wire/cable and use it in single runs with very little solder on the joints (just enough to make the connection).
As far as the small size of the wire goes 26 guage is heavy enough for amplifier power up to 100, or so, watts per channel, from my experience, and friends have used it with up to 350 watts per channel. Try 24 guage if you wish to shift the balance more towards the bottom end - not really more bass, just a slighty different balance. As far as larger wire producing more bass, well, I do not agree with this as my 26 guage speaker cable has just as much bass (or more) than any of the heavy commercial cables that I have used in the past.
You need not spend a lot of money on your experimental cable as dealers (such as Michael Percy Audio) carry high quality stock in bulk (you order by the foot). I have not found a source for ready constructed solid core with "heavy" Teflon tubing so other than the 47 Labs OTA cable that I use, I also make my own from bare wire and thick Teflon tubing. I was @ a Fry's Electronics last week (picking up a new fan for our computer) and found some fairly heavy Teflon tubing that I will be experiementing with (it was less than 10 cents/foot - with 30-40 feet on a roll, depending on the guage).
Anyway good luck (lot's of good advice already in the thread) and as mentioned be prepared to undo what you have done in the event that you do not care for the sound. Kind of odd that such short runs of cable can make such a difference, but they do and will change the sound. Maybe just play around with the wire first before experimenting with caps?