How to Actually use Headhunters to Land on the Buyside?

Hi all, 
I made this throwaway account to basically ask generic questions so that it might lead to a better discussion and potentially help others that have my same questions but may not want to post. Given the WFH policy that's gone on for much of the year and the fact that it's actually easier now to interview and chat with HHs given that you dont have to sneak out of the office it got me thinking: How exactly am i supposed to go through the ballad of the headhunters?
I am currrently in a European Office (major office, not "our-secretary-helps-us-with-DCFs-when-we're-shortstaffed" type of office) for an American BB at the Analyst 2 level (my division and desk are not important to the discussion so i won't reveal them), and I have a target school MIF, so I probably fit the demographic of most users here very well. Maybe I'm an idiot, but so far nobody has sat me down and told me what to do and say to these people so that I can get a decent position. It always seems like its "if they think you're interesting for a positon they have, they'll get in touch" but that seems stupid to me so it got me wondering.
There are quite a few posts on WSO that ask for recommendations on who the headhunters are and what firms they have relationships with but i couldn't find anything on how to actually be proactive in your approach, and make the most of the process, especially for the European scene. The questions that come to mind which I, and I am sure many others too, would like to have answered are the following:1. Can I reach out to them?
Can i just look some of them up and basically tell them: "Hi, I am not currently looking to jump ship but I would like to catch up with you in around 6 months to chat about what type of opportunities I would be eligible for" or should I just wait for when I am actually ready to leave? In other words, does it pay off to actually try and foster a relationship with these people? If so, how often do i check in? Once a quarter/6 months/year? do I need to give them job updates (I worked on deal X/executed trades for Y amount/Institutional investor ranked my analyst #1)? Do i target the more senior HHs or the younger guys?

2. How specific do I need to be when telling them what I am looking for?
Are we talking specifying an asset class/product (eg. GEM HY) or type of fund (fundamental/quant/HF/PE/LO/FO)? I understand that you want to show them that you've thought it over and are not the wishy-washy type that will waste their time but I would also prefer to get a good shortlist of broader positions I can really take my pick from. Stupid example but I don't want to tell them "KKR or bust" if they also speak to small shops that offer better learning experiences

3. Pay: do i need to tell them what I make and how much I want? I presume they'd have an idea, especially for the former, correct?

4. Do they ACTUALLY know who the good analysts are in our teams or is it just BS?
From what i've read here the answer would be yes, but I have a hard time believing that. Why would associates, let alone staffers/directors, want to share that info with someone who could steal their juniors from them? 

5. If i want to move cities, do I need to get in touch with HHs there? 
Will they give me the light of day provided I come from a strong bank and know the language or is it basically a waste of time considering regardless of whether i want to do London->Paris or Frankfurt -> Milan there is a myriad of readily qualified applicants in local markets already, meaning I'll never get looks?

6. Re-using the same head hunter
Imagine I landed my BB job through a HH and now I want to leave. Given the HH will have a relationship with my bank is it stupid to ask him/her for help because he'll sell me out or is it smart because they know i can be put in front of people and walk away with offers?
Cheers, hopefully this leads to some quality discussion