Becoming a lord/lady doesnt mean anything right now, really. If youre the only contender for the throne that actually holds the mead hall, then you are defacto in charge. Other groups can always claim rulership, but having access to the mead hall is the most important part of being respected as the ruling government. Making a claim as lord of a site will revoke your status as a hearthperson to other lords, though.
Hearthpeople are odd at the moment. Sometimes theyll migrate to your site but i think the majority remain where their home location is. Sometimes recruits you never asked to join and never met just appear in the mead hall and if asked will claim to be your hearthperson. You cant reclaim any sites except dwarven forts, and you cant really stake a claim on dwarven forts as an adventurer because they technically dont haave a designated mead hall or governing throne area in either generated or player forts. Im not sure if cities will repopulate or not, but things like temples and taverns in abandoned cities will be restored after making your claim, even if nobodys there.
Im not totally sure what your lord/lady actually wants you to do, but i think harassing people and demanding they yield and give you items should work. You can start fistfights, but keep in mind murder is murder as far as your lord will be concerned.