Alex has no set home or place to live at, being on the road most of the time, they have set up multiple dens to live in so that they won't get found easily, changing location after a big job with the help of some smugglers. As for why they do this, it's because of the ties to their past, being a lab experiment created by the DSP, that they are forced to move every so often and not live the normal life they probably should have deserved by now. They are now an outsider, a joker in the deck of playing cards and everyone is looking for her, not giving her a chance to breathe.
Mostly on stuff like supplies or other tools that may be useful in the profession of an assassin, Alex doesn't really have to use it on stuff like food and water because of their body not needing it, they can still eat and drink but don't need it like normal humans do so they don't buy anything for themselves, only stuff like coffee to push back the need for sleep, basically hibernating and saving up excess sunlight for the nighttime whenever their body can so it can last for that night. Stuff like their criminal contacts and smuggling partners also get a cut of money to stick with her of course.
The Ambition to stop the abuse of fellow gems by putting a permeant end to the gem trade & industry, is more of a goal for the sake of a goal, they needed something to go after in their life and seeing gems as her brothers and sisters helped her form a full direction on where to go, not to mention, the real reason why they want to stop the gem trade is so that DSP and other organizations will not create more gemoids like her anymore, knowing her kind would just be abused and made into a commodity, like she was when 'being taken care of' by the DSP in one of their larger and complex research labs in the world.