I've never seen a set of carbs that have been knackered from just standing, usually it's someone messing with them! So don't give up hope yet.
The vaccumn pipe is a small pipe that runs from cylinder #1 (bottom left)
They can be a pain in the arse to start anyway, any hint of throttle on a cold engine will kill it straight away. You want to be trying full choke no throttle till it fires, waiting a minute between each 10 second go.
I'm presuming you've checked you have a spark.
As its running then dying....
Firstly: Check the choke mechinism, it's a series of rods that pull out a plunger on each carb, make sure this releases properly.
Secondly: Float heights could be wrong / pilot screws out.
Thirdly: Air filter / Air Box ok?
While its apart anyway your probably worthwhile giving it a good seeing to.
Whip the rocket covers off and check / adjust the valve timing. (you'll need to strip more things out of the way but it really is worth doing / checking.
Oil / Air Filter / Plug change.
If all of that sounds too much, just overhaul the carbs.. they can be a bit of pain to get on / off the rubbers until your 'used' to it. However:
Strip and clean the carbs, theres only 3 screws on the bottom of each float bowl, the gaskets are rubber so are usually re-usable.
Blast everything with carb cleaner, blow compressed air through all of the jets (after taking them out and checking their ok) make sure all the linkages, jets should be 115's allround and 35 pilot jets unless someones been messing. The rear emulsion tubes have an extra hole in them.
Check the sliders move ok and theirs no tears in any of the diaphrams (4 screws on the top of each carb) you'll also see the choke mechinisim when you remove the diaphram, blast all this with carb cleaner too and move the rods in and out to make sure their free.
Check float pins, float heigh and pilot screws and then re-assemble.
All carb info on my website
http://www.jacksons-lair.co.uk/specs.html" onclick="window.open(this.href);return false;
hope thats of some help.
PS: with your burning connector.. make sure your running a decent battery, if thats been burning the reg-rec's been throwing off too much heat, usually caused by a failing battery unable to hold a charge, if you dont' catch it soon enough it burns out the reg/rec, then burns out the altenator so you have to replace reg/rec / altenator and battery! for the sake of a good battery in the first place.