The PRIME position by-passes the vacuum-tap and fuel will flow without the engine running (and creating vacuum in the no. 2 inlet tract which is connected to the petcock and opens the tap). You can leave the tap in PRIME until it starts, or if you have a fuel overflow problem (as you mention below) you can turn it to ON as soon as the carb bowls are full.
The choke should be put to full BEFORE you hit the starter button. DO NOT open the throttle with the choke applied. The 'choke' is actually a separate cold enrichment circuit (a mini carburetor) that requires full vacuum from the cranking engine to work. If you open the throttle you destroy the vacuum to the enrichment circuit and it doesn't flow fuel.
Run a long hose to the fuel tap outlet into a suitable container, run another to hose the vacuum spigot. Test the tap. PRIME - fuel should flow. ON - no fuel unless you suck on the vacuum hose, stop sucking on the vacuum hose the fuel should stop flowing. Repeat this in the RES position. You should get the same result. If you get a mouthful of gasoline, the diaphragm in the tap is shot.
First positively identify the source of the leak. Is it from a vent hose? Is it from the carb bowl gasket or is it from the junction pipes (fuel gallery) that connect no.3 (RH inner) and no.4 (RH outer) carbs? If its the pipe, its likely to be the o-rings on the junction. If it from an hose or weeping around the bowl gasket, its likely to be the needle valve -either worn or stuck open. Fixed by replacing or cleaning.
Starter draws a lot of current hence the thicker gauge wire. Prolonged cranking can cause heating on old positive starter leads. If the lead gets hot without cranking it will be shorting somewhere and that should be fairly obvious.
If its the standard airbox some wrestling will be required. In broad terms, the procedure is to remove the tank and anything that will get in the way. Disconnect throttle and choke cables. Loosen the hose clamp between the air filter box and the plenum chamber that contains the 4 hoses feed air to the carbs. This 'should' allow the plenum chamber to slide backwards (alternatively you could remove the air filter and box). Unbolt the plenum chamber mounts and loosen (all the way) the hose clamps between the plenum hoses and the carbs and pull the plenum box backwards away and off the carb set bell-mouths to gain the meagre clearance afforded. Next is to loosen all the way the inlet hose clamps between the head and the carbs. The rubber hoses on both sides of the carbs are likely to be colder and harder that a Nuns heart on a Sunday, so trying to make everything rubber a little more pliable with a hairdryer (don't use a hot air gun unless it can be turned down very low - these are for burning and melting things). Next you need to pull the carb set from the inlet hoses. This is likely to not to be an easy task after all these years.
Once you get the carbs free from the inlet hoses, you will discover there isn't enough room between the front and rear hoses easily pull the carbs out the side of the bike. They will come out, but it takes some tilting of the carb set and deformation of hoses (hence the hairdryer) to get them out - and back in. You will be left sweating, but if you take it methodically and look where the carbs are catching on the hoses you will eventually get them out. Just the effort of doing this, should ensure that you do a surgical cleaning job on the carbs. Only a sado-masochist finds joy in pulling carbs from a bike with a standard air box. On the plus side it does get easier the more often you do it. Just on other tip, the bracket that holds the throttle and the one that holds the choke cable can be removed easily as they are only held in with one or two screws that hold the diaphragm covers on. Removing them before pulling the carbs gives you more clearance for extraction and refit.