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Sectionals say to contact the approach center of a class C airspace within 20 miles. Is this a suggestion or a rule? 
What is the maximum distance to contact them via radio, other than signal limitations, if one exists?
What is the maximum distance to receive radar service from class C?
Note that class C radar service and flight following from an ATC center are not the same thing as it pertains to my question.
There is no such thing as "Class C radar service". "Class C airspace" and "radar service" are mostly orthogonal. "Flight Following" is just shorthand for getting radar service when you're not required to.
Terminal areas around a Class C airport generally comprise a 5-mile "inner ring", 10-mile "outer ring" and 20-mile "outer area". Approach is guaranteed to control (at least) that much airspace, regardless of class. Within the airspace they control, they provide (roughly) the same radar services regardless of whether you're in Class C or Class E at any given moment. The only practical difference is that in Class E, you're not required to talk to them, whereas in Class C, you are.
The reason the box says "CTC KNOXVILLE APP WITHIN 20NM" is that 20 miles guarantees you're within the "outer area" and thus Approach's airspace. If you contacted them at 30 miles out, you may be Center's airspace, in which case Approach would need to redirect you to Center, and then Center would hand you back to Approach a few minutes later. That's a waste of both your time and the controllers' time, so the chart reminds you of the minimum range to help you avoid that.
There is no theoretical maximum, and many Approach facilities do in fact control more (sometimes far more) airspace than the minimum. However, the FAA does not publish the actual sector boundaries, so the best practice is to assume no more than the minimums—or get Flight Following for your entire trip, in which case you don't have to worry about it at all.
Is this a suggestion or a rule?
Rule, if you are VFR and landing there. If IFR, or VFR with flight following, you will be told when to contact approach when arriving from elsewhere, and approach will hand you off to tower. If just passing by, I would at least monitor them to help with situational awareness of traffic in the area.
What is the maximum distance to contact them via radio, other than signal limitations, if one exists?
I'd be contacting them within at least 10 miles, before you enter the Class C airspace. You can try them as far out as you can receive the ATIS. For example, I was flying VFR into Binghamton, NY (BGM) and called them out about 20 miles after picking up the ATIS - and they told to enter right base for the active runway and I was cleared to land! Slow day I guess, not much other traffic.
What is the maximum distance to receive radar service from class C?
Usually 10 miles. If VFR and departing the area, they'll often ask you to advise them when you are leaving the Class C. If flight following was requested, they can hand you off to approach quite soon, like 500 to 1000 ft AGL.
