Les paul making wierd noises

Or a bad solder connection on the ground wire on the jack, or shell/sheild on the guitar.

Here is the list you seek.
http://www.gibson.com/en-us/Support/TechSupport/Wiring Diagrams/

This is a common problem... Make sure your amp is plugged into a three wire grounded receptacle. Keep the guitar away from magnetic fields of transformers in amps and effects units. Make sure you have a high quality guitar cable.

Common problem is broken wire in the cavity where the pots and wiring are. Open it to investigate. Less common problem is a pot rotates and one of the terminals touches the side of the hollowed out cavity that is coated with conductive paint and shorts to it. Last problem is a broken jack. ALWAYS loop the guitar cable through either your belt or the guitar strap to avoid the twisting wear of the plug on the guitar jack that will then get noisy from wear.

The half-way cord is the clue. Read up on unbalanced and balanced lines and TRS connectors.
One of these requires TRS connections and the signal is being carried on the RING of the TIP-RING-SLEEVE connector. Plugging in half-way the TIP touches the ring pickup contact. The emgs one MAY use TRS type cables and there may be a reason the signal and hot of these are reversed. If I could find schematics on all this equipment I could unravel the exact cause. You could make an adapter with a Stereo 1/4 inch plug and a 1/4 inch jack by wiring it to swap the tip and ring between them to use for connecting to the emgs guitar. ALSO verify you are using the SAME type guitar cable for both.OFTEN a TRS cable will NOT turn on the electronics when plugged into a guitar. Use an unbalanced instrument cable unless you have the XLR type output. Use of an adapter from the XLR type cable into the Boss will usually not work.

Check the power supply. Dead batteries or a failing power adapter can cause this.

OK... Most of the guitars have a PAINTED on shielding in the well with the electronics. Sometimes the electrical connection to this is broken causing hum. Sometimes the connection is by way of the potentiometer shaft nuts and they spin and break the connection. This paint is conductive and often black know as aquadag. Use an ohmeter to verify connection.

change the cable wire connect the negative line as a ground

Sometimes the sound can feedback onto the strings of the guitar effectively sustaining.

Also check it with reverb off. Sometimes the effects loop and create sustain.

I take it you have made sure that both the cable to the distortion box from the guitar and the cable to the amp are both good. From the description you give, it sounds like there is no signal getting to the pedal. If the cable is good, then the next likely suspect would be the input jack on the fuzz box. Try wiggling it a little with the guitar cable in place to see if it is loose. Some input jacks are unbelievably cheap and flimsy. A crackling sound (static like) is almost always a bad connection. If it was the output jack, I would think that the guitar signal would come through a least a little bit, but you might want to try wiggling that one, too, if the input jack isn't the culprit.

Another possibility to check is the battery, but this doesn't seem as likely. Some effects units devour batteries, especially units that use 9V batteries. I have a multi effect pedal that kills a 9V in about an hour.

This is a very common problem that I have repaired for many friends. Sometimes the guitar jack becomes loose and gets rotated to the point where the wire breaks and/or touches the cord plug when it's inserted into the jack. You need to remove the access plate/panel at the back of the guitar body and see if the wire is broke or twisted. You may only have to loosen the jack retaining nut and rotate the jack to get the twist out or you may have to remove the jack and resolder the wire back in place. If its broke make sure to solder back on the proper terminal--the one with solder already on it and not the one that the cord plug touches when its inserted. Retighten the jack nut by firmly holding the jack from access side so it doesn't rotate again and cause that undesirable STATIC sound. Hope this helps.