Out of curiosity, are those in the 02 code block for you, too? If not, do they at least start with 0?
Yep, 02## are our local ARTS codes. If you want to handoff someone to the Center, you have to give them one from the FDIO.
The FAA Private Pilot Written Exam asks a question about when it is most appropriate to request flight following. At a towered airport, it is when on the ground. The idea is so the controller can, in fact, type in a VFR flight plan for the radar controller. Most towers have that capability nowadays, but you'll know if they don't -- they'll simply tell you what frequency to use to request VFR FF.
I figured something was up with that, because at SUS, we had people request it with their initial call. In Class B, you're going to get it at least through the airspace, and we'll ask you if you want FF with STL approach before giving a frequency change. The ones at smaller facilities that don't enter the flight plan are probably just being lazy. We also had some that just said that since there was no directive saying they had to enter a flight plan, they wouldn't do it.