Call a plumber. Remove exterior cladding, insulation etc, down to where the offending part is and replace it. Then restore the wall. There is often a choice between going in from inside or outside. I usualy opt for the outside in approach, but both can work.

Is the leaky faucet anywhere near where you have the other musty/dampness problems? Perhaps fixing it will solve two problems. If you encounter longstanding wetness or even just dampness, remove adjacent parts and replace. Do not reclose and hope it will dry out on its own. It won't and you'll be setting yourself up for serious mold problems. Bad for you and terrible for resale.

Oh, and have the plumber install, at least, a frost-free type faucet (though still can't leave a hose attached in the winter) or, better yet, install a drain cock on the supply line on the inside in the basement so you can easily drain the water back out during any freezing weather. Will cost a bit more but you'll be so grateful to have it easy to do!

Good luck.

Molly

Easier to do now before it breaks and before winter.

If you have any desire to learn simple plumbing repairs, this is a fairly easy project to complete on a DIY basis.

I'd also suggest one of those foam caps for the faucet head for the winter. They only cost a few bucks and will save you grief should you not get every single drop of water out.

You can try and replace the packing (Teflon packing works well) or install a 'freeze proof' faucet.
If you decide to install a freeze proof there is a little 'trick'. Use a threaded fitting at the back of the faucet and screw it in. This allows the faucet body to be removed from the wall without tearing into things to unsolder the joint.
'Freeze proof' faucets can still fail if the hose is left connected or they are not pitched downwards to drain completely. Unscrewing the body is a much simpler repair than opening the wall up again, particularly in brick walls.

That sounds like a lazy GC. This should not be difficult. I just replaced the original outside faucets on my 1952 house and all it took was some Liquid Wrench and a medium size pipe wrench. Total time per faucet: 1 minute to squirt on the Liquid Wrench, 20 minutes to let it work, 1 minute to muscle off the old faucet, and another minute to apply teflon tape and screw on the new one.

It occurs to me that your faucet is suffering from either a bad seal or internal corrosion associated with age. There is no damage due to freezing weather, and the word "frozen" used in your initial post refers to a threaded part that is stuck to another threaded part. If that is the case, then, I agree, repeated application of penetrating oil to the threads should eventually result in success at removing the faucet, and threading on a new one. It can take penetrating oil a month or more to reach the affected threads.

wow, GC wants to leave a faucet that is causing damage to home alone? get it fixed. call a different contractor, or like said earlier, by others, it you can do it. if it is covered in paint, remove the paint, any auto parts or walmart store should have numerous paint remover chems, then disassemble the faucet take the parts to a pluming center, and find new gasket/seal and reassemble.
turn off water first!
I also like the freeze proof type of faucet. the body is usually 8'12" long, and it is thicker than a standard copper pipe, but as long as the hose is disconnected, no water will be exposed to cold in the faucet.
beware of galvanized piping in the house, if you need to dissasemble faucet from plumbing in house, and it is galvanized pipe, you could indeed be in for more trouble than it is worth. galvanized pipes will deteriorate from the inside out, and when you touch them with wrench, pop goes the weesel! total pluming replacement can sometimes be needed! thousands of dollars.
then again, if I had galvanized pipe still in my old house, I would want it all removed! Water pressure and supply problems GONE!