Agnes: now her being here may surprise you, but during arc 3 she wasn't doing too well. roaming around the country looking for any clues about Ryan's whereabouts means she spent a lot of time without a house or a steady income. she had to sleep outside lots of times, and also steal food another bunch of times. for her, Ryan was priority number one, and anything else was an obstacle between her and him. she kept going on until she completely burnt herself out both physically and mentally, and gave up. then the process of settling into a normal life was another big challenge she had to overcome. it was her personal punishment for failing her mission. for failing him. she firmly believed she deserved every single second of misery she went through
Rask'r: he had some stability during his childhood, then the werewolf pack that adopted him taught him to survive in the wild, so when circumstances separated him from the pack, he was ready to continue on his own. to him, being a vagabond is just how life is. he hunts for himself, he seeks shelter for himself, he travels wherever he pleases for himself. being homeless is freedom to him. having to get in cities to trade, get and fix gear and find mercenary jobs is what he actually dislikes. walled cities make him feel caged in, and they're constantly buzzing with activity. how noisy!
Raz: started off kind of like Rask'r, but things went very wrong for him too soon. his foster father disappeared when he was just a toddler, not having enough time to teach him about lycanthropy or survival. he ended up being unofficially adopted by an elderly couple once he appeared all bloodied inside their chicken coup, shaking in fear. things went fine for a while with them, until a specially bad full moon. he accidentally hurt one of them during it. when he saw what he did the next day, he realised that despite how the grandparents were doing their best to care for him, they were too frail to handle him. he was too dangerous for them. so he ran away. he kept himself fed by hunting while transformed, he didn't have any other means to get food. now i'm doubting if he gets "captured" and put in an orphanage during his late teenage years because being a monster while being confined to a building inside a city is a very bad situation and i really doubt he could manage to survive that without outing himself as a werewolf and not hurting anyone. either way, he escapes and ends up officially homeless by the time he reaches his 20s. there's a portion of his life where he lives with a colony of ratfolk in a city's sewers; it's also when he meets Eva and she offers him to stay at her place for a while while he tries to save money for moving out of the city, since the city monster hunters are tracking him. then the science facility fiasco happens and then he's forced to go full homeless, escaping the law. as for coping, it changed over the years. during his childhood it was very rough since he didn't really have anyone to teach him how to take care of himself. as he grew up, life itself was his teacher. tough love, as they call it, although the ratfolk were very kind to him. the Big Thing that keeps him from being happy is not being able to find balance between him and his beast form. he doesn't know what to do with it. he copes by separating it from himself, he gives it a name, a personality of its own, ANYTHING to refuse to believe it's just another part of himself. it takes him a very long time to realise his wrong approach to his situation, but he eventually does it, thankfully
Donovan: after being revived, he had no problem with having a roof atop his head and food on the plate. quite the opposite: that was his pirating prime era. it was much, much later when things got tough for him. luckily he was at a point where he didn't need to eat anymore and sleeping was optional, but it still opposed a problem to him. the solution to his problems? a van. he got one (don't ask how, he's a pirate, take a guess) and turned it into his portable house. quite the upgrade, since now travelling by sea is unviable given the technological advances. as for coping, his apathy gives him an easy way to deal with it. he's just drifting around, with no real aim or purpose. he stays comatose for most of his time, and when he's awake he's mostly dissociating
Lee: technically being a stray is being homeless, right? regular stray animal struggles for him: finding food and shelter is his daily life. as an anim, he doesn't really need to cope because of it, but it does cause him stress and makes him skittish
Vreytus: he was technically homeless before he became sapient, but the same way a stray dog is. living driven by instinct until a wizard "adopted" him and unlocked his sapience. after that, not much changed. chimeralmas don't need to sleep or eat (doesn't mean that they can't tho), so not having access to food or shelter doesn't really matter to him
Bug: same as Vreytus, it was living its own life as a wild animal until it got captured by shady researchers. having to deal with being experimented on and being exposed to humans and other sapient creatures unlocked its own sapience, eventually leading to its escape. since it doesn't really have a place to go back to, it just roams around, going wherever its curiosity takes it. no food/shelter still ain't a problem to it, but it still makes it look shoddy when it disguises itself as human to enter villages. they legit look like an escaped prisoner/slave with their ragged clothes, buzzcut and being short and underweight. not that it bothers them, of course. they couldn't care less about their appearence
Chad & Lewis: they have to survive all the way from HL1 to HL2 on their own, in a country that's being ransacked, polluted and ultimately destroyed by aliens. being together makes things easier, but they still have a hard time finding shelter and food. Chad is smart enough to successfully hunt and Lewis knows enough about flora to make sure they don't eat anything poisonous. during the first years after the Seven Hour War, their focus and one motive to keep them going is to not be captured by the Combine and simply to survive until they eventually learn about a resistance forming in Europe, but there's an entire ocean between them... still, the thought of humans actually fighting back against the Combine encourages them enough to travel to the east coast and find a way to cross the ocean to the other continent. which they luckily do! bless Aperture Science and their nonsensical inventions. once in Europe, it's true that there's more Combine to hide from, but also there's more rebel outposts to take shelter in. they jump from one another until they arrive at White Forest, and then they make themselves useful there