The crack in the exhaust manifold is likely allowing some oxygen from the air to enter the manifold between pulses and this affects the oxygen sensor signal. The erratic signal is likely throwing the code.
It should be pretty inexpensive to replace the manifold with one from a salvage yard though the rusted bolts on the outlet of the manifold may give you some trouble.
I'd replace the manifold before you replace the o2 sensor.
If it is like my civic, there is a joint between the two- so you won't have to replace the cat. I had to twist off the bolts between the cat and manifold- unless the cat side is threaded, you can just replace them with cheap bolts and nuts from the hardware store.
Nope. If it's the D16Y7 like my DX, the manifold and cat are one piece. Look around for aftermarket parts as OEM will be expensive and have the same flaw. Junkyards will have cracked ones as well. Mine has a crack but it's not horrible (yet).
I heard from someone who did his own swap to be careful with the O2 sensor - easy to break off.
I think they are cast iron, mostly. Not impossible to weld, just not easy. Parts that have wide temperature operating ranges (like exhaust manifolds, woodstoves, etc) tend to have a high tendency to have repeat cracking. It's kind of a crap shoot. Most welders hate welding on it.
Sometimes you get lucky and things are just ducky.