Go to the source. You will find the relevant code online.

Lot of concrete there! I don't know if you can use it on old concrete but my son expanded his front steps greatly to 8 ft long & 4 ft. wide & somehow made a rounded edge on them. H e had to wait awhile to put the finish on but it was called a "stain" & it is really beautiful, looks like a walkway in Italy, marbled old world looking with deeper looking "veins" for an aged look. Everyone looks them. I asked him where he got the idea & he said he had seen directions for using the "stain" on the Internet & just did that. I'm sure it comes in colors. His is a vanilla/gold Venice look. Not sure what you would look for but try "concrete stains" & see what comes up, he watched a video & then just did it. He did his patio also. You have some darkening on some of the cement so might be wise to power wash or clean it before doing anything to it. You would want a color to bring out colors of the flagstone. Hope you can find it. I can ask him what it is called & if it can be used on old cement/concrete.

The vinyl covered ones you are referring to are control joints rather than expansion joints. The expansion joints run full depth and are filled with mastic over a foam filler strip. Control joints are only about 1/2" deep and create a line along which the concrete can crack without it becoming unsightly. I don't know why they use the vinyl strips over control joints on pool decks, control joints can be just a simple tooled joint or sawcut joint as you see in sidewalks. I guess they think it has a more finished look. If you don't like them I'm sure you can ask the PB to just do a tooled joint and not use the strips.

Concrete exerts a lot of force on forms. It would be very easy to get that kind of distortion simply by temporarily having a little more concrete in one section than another as the pour was being done. I agree it's esthetically unattractive, but I don't think it will compromise the integrity of the driveway.