Your father would be proud!

My 2 cents:

Wheels: It's strange to me that the wheels flex under pressure. Unless you are a beast, I wouldn't expect there to be much flex. Those appear to be steel wheels, so they should be strong. It could be that the spokes are under-tensioned. You could take it in to the local bike shop and have them true your wheel, but it's a great skill to have if you don't already know how to true a wheel. A truing stand is ideal, but truing your wheels while on your bike using the brake pads (or zip ties to the seat stays) is perfectly adequate. Here's a brief tutorial: http://www.parktool.com/blog/repair-...and-rim-truing

You may indeed wish to upgrade the wheels to something with alloy rims. Steel rims are terrible in terms of braking performance. New Kool-Stop pads will help as well, but it may never stop very well with steel rims--especially in the rain. Keep in mind those are 27" wheels. There are more options in terms of both rims and especially tires if you switch to (slightly smaller) 700c. However, will likely not have enough reach with the brakes if you do. Personally, I'd look for 27" alloy wheels, Kool Stop pads, and live with limited tire choices (Panaracer Paselas are a good, inexpensive option).

Headset: I would a get this resolved sooner than later. Again, the Park Tool website has some helpful tutorials if necessary. Riding on a loose headset not only affects performance, but will also make the headset prone to pitting.

Front Derailleur: What do you mean it's stuck? Could it be the shift lever?

I have a question for you: Do you see a Serial Number on your frame? If so, what and where is it? I have a rattle-canned frame that is very similar to your and I've speculated that it might be a Falcon. Might help me confirm if mine is also a Falcon.