Unfortunately I do not have the shop manual for you machine, Or I could be more specific, but in general the relays on your tractor are probably similar to mine, in that they are small square devices on the firewall of the tractor, they usually consist of a coil connection so that when current is passed thru the coil a magnetic field is generated that causes the armature, the moving part of the relay to move and make a set of high current connections. Think of the relay as high current switch that instead of being actuated by your hand is actuated by the flow of electrical current. The start relay would be one example of this and the glow plug relay another. But instead of focusing of relays, lets focus on the basics of what we know and don't know. From what you have told us, it would seem that the battery is adequate to operate the tractor, yet the starter does not spin, and as a side note the dashboard light do not seem to operate properly, and we hear some buzzing noises like a relay perhaps lacks sufficient current to pull the armature in and keep it energized. Since we seem to have eliminated the battery as a source of these voltage drops, the problem still sounds to me like high resistance in the primary cables or their connections. I am not sure how your tractor operates but in my tractor, when you turn the key on it goes thru a glow plug cycle, controlled by the glow plug timer circuitry and a large glow plug relay. This draws a large amount of current out of the battery to heat the glow plugs during the cycle, as long as the coolant temp is lower than 86 degrees. If for any reason there is a high resistance in this path the voltage to the rest of the tractor would drop tremendously due to the large amount of current potentially drawn by the glow plugs. Can you measure the voltage on the glow plug rail?, Can you measure the voltage on the lights? Do they even turn on, and if so are they dim? Do the flashers work? and are they dim? They should work with the key off.. Again, I don't have your schematic and I am sure it would be somewhat different than mine, but perhaps you could perform some of these tests and get back to us. I am still betting on high resistance wires or connections though..
James K0UA