Keep in mind I am still somewhat of a beginner. The way I understand what you wrote is that you would use the 60 Hz mains as a clock that you would then use a counter to keep track of time. When it reached 12000 it would stop. This is a great solution for this part of the equation.
You can get close, but you do need a bit more than a single 4020 divider.
If you have 60Hz mains, that's /720000, and most long counters are binary.
Thus log2(720000) is 19.45763738 binary bits
The closest binary taps for your target of 200 are
2^19/60/60 = 145.635 minutes
2^20/60/60 = 291.271 minutes
If we assume you have outputs available on all binary counters, you can wait a bit longer, for more than one output to be HIGH.
thus
(2^19+2^17+2^16)/60/60 = 200.248 minutes - ie an AND gate combining counter taps at 19,17,16 will first pulse high at 200.48 minutes.
You wanted some adjustment, so you could for example pick taps at 19 and 17..14 for 16 choices (4 switches or jumpers), next longest is 204.8 mins or a ~2.2% step size.
More taps give finer digital settings, eg 5 taps/switches or jumpers is ~ 1.1% step size.
Or, if you like analog, you can use weighted resistors and a RRIO comparator and trim pot, to 'pick a step'.