No as standard equipment on the coupe. This is another item that gets debated, but the general rule is wagons and V8 cars regardless of body style had them. Some say SS cars did, but I have had three and none did. My 63 wagon also did not have a sway bar, but it could have been removed.
Yeah, it is not a drastic improvement like on other model cars. The really big ones (inch and larger) can have rubbing issues on the strut rods. Also, your lower control arms likely will not have the cups to mount the end link bushings in. The bar can still be mounted, I have done it, but it allows movement in the assembly and premature bushing wear. Things the Chevy II does not need more of. Solutions for this are swapping in lower arms with the cups or welding something in to act as a cup.
My 64ss didn’t have the sway bar or the crossmember when I got it. I pulled the sway bar and crossmember from the front clip of my 63ss.. I believe this is a 7/8” bar. You can get larger diameter bars from Afco, CPP, and probably other suppliers..