Several things going on:
To start with, your wiring is "correct", but old-school and requires the thermostat to steal power to keep the battery charged. That's not ideal, but is straightforward fix; some people don't have issues leaving it as-is.
For the fan: Temporarily jumper the R and G terminals in the furnace or remove the thermostat face and jumper at the thermostat end. If the fan starts, that's a good sign: simulating the thermostat manually starting the fan will let you know if the timing function on the thermostat will be able to run the fan without the heat or cool also being "on". If it does not - there's likely an issue with the fan's limit switch. (I'm assuming the G20 has one - I'd need to see the controller circuit). Some circuits don't allow manual (timed) fan operation unless a switch is set properly on the furnaces controller.
Wires going to the condenser: The colors going to the condenser technically don't matter (one wire in of any color and one wire out of any color is fine). It's a common "old" practice to splice the condenser between the thermostat Y and furnace T (transformer) or C (common). Ideally, it would be yellow "in" (Y) and blue (or black) "out" (T or C)
Blue wire: To stop power stealing, connect the blue wire to the thermostat C and the furnace T. That keeps the thermostat charged and prevents some cycling issues.
Removing the condenser splice (an option) If you can wire the thermostat Y to the furnace Y; and the condenser directly to the furnace Y and T, that gets the thermostat out of the middle of the condenser circuit. To do this: remove the splices to the condenser and splice the thermostat Y to the Y (in the same cable) going to the furnace. Get a piece of 2-wire or greater wire count thermostat cable and connect the red wire to the furnace T and yellow wire to the furnace Y. The other ends of the wires get spliced to the condenser R and Y you disconnected, respectively. Using the same colors makes it easier on the next person that has to work on it.
Hope this helps.
