Firstly is it okay to have more than 1 Galcop mission open?
Theoretically, yes. Although, TBH, I haven't tried every combination of missions to see if there might be negative interactions, so you might have come across a scenario I need to cater for.
So I start the first of my gun runner missions (thinking that it will be very similar to the slave release mission I had just completed where I just had to use broadcast coms to request ships to drop any slaves they were carrying
In this case, no, the gun runner and drug dealer missions don't have a Broadcast comms method of asking. I suggest using your laser to request the cargo! Seriously, I'd also recommend the Manifest Scanner, that can scan ships cargo and lets you know if the ship has some of what you're looking for.
Anyhow after undertaking this mission in Esbete (galaxy 2) I noticed that there was the same type of mission in nearby Atorat
The system should allow for you to have two of the same type of mission for different systems, but not for the same system. If you end up with 2 in the same system, it's a bug.
I have been jumping using Torus to Sun oxp and periodically checking both my Adv Space Compass and also Range Finder MFD for coms relays (not sure if it would show up here or not).
(As an aside, Torus To Sun probably doesn't increase your speed over what the core game now does for you, since v1.82) Range Finder may or may not help in this instance, depending on what's waiting for you at the destination. However, you should have been able to pick up the signal soon after passing the wp beacon. I'll do some tests at my end and see what I can find. In the meantime, if you can send me a copy of your save file that could help me quite a bit.
Thanks for all your help with this!