I think we will still need the pictures Laurie, and in any case, without them this thread will be useless to other members. The main problem is that the wire colours you have quoted do not match the Rover Rancher manual on this site. Perhaps there is a model mismatch, or your mower has been rewired. It would be a bad idea to proceed without either finding a manual that matches your wires, or tracing your wires with the correct manual in front of you. Expensive damage is likely if you make the wrong connections.
According to the manual, the original wiring of the mower itself (not the B&S engine) has these wires going to the engine:
a ground wire from the starter solenoid negative terminal to engine ground (colour not specified);
a red wire from the starter solenoid to the starter motor (to activate the starter when you operate the key switch);
another red wire from the safety circuit that goes to the magneto (this is the only connection to the magneto, and according to the safety circuit diagram it is the ignition kill wire - in other words it is grounded when you need the engine to stop, and floating when you need the engine to go);
a green/yellow wire from the key switch terminal L that goes to "the engine" according to the manual. Since it is the only wire that comes from the engine and goes to the key switch it seems to be the battery charging wire from the alternator; and finally
a black/white wire from the headlight switch that goes to the alternator.
As I said, some of your colours don't match these, so you need to subscribe to the manuals section, download the manual, and trace your circuits. In my experience most of the people who come here with ride-on mower electrical problems have mowers that have had their wiring butchered. The only proper cure for this is to rewire the mower in line with the manual. This is not a difficult job, so long as you have the manual.
With regard to the float bowl solenoid on your new engine, all you need is a wire from the contact on your key switch which becomes live when you switch on the engine (this seems to be terminal L, or in other words the green/yellow wire) to the positive contact on the fuel solenoid, while ensuring that the negative contact on the fuel solenoid is grounded. Most likely the black wire will be the ground and the grey wire the positive, but you can check by briefly connecting it that way and listening for a click. Now to test it you will need a voltmeter or test light: with the green/yellow wire disconnected at the engine end, turn the key switch to engine on and check that it suddenly moves to plus 12V. Then do the same when you turn the key switch to start: it should go to plus 12V then as well. Finally, turn the key switch off and verify that the green yellow wire loses its positive voltage.
