I think you need to talk to the rest of the family about what happens going forward. He's a teenager whose mum has just died, so he does need support, but aimless ricocheting from his siblings' houses, all across the country, isn't going to help him long term. He needs to settle somewhere and start moving forward.
I think he needs to think about what he wants for the future, and where he wants to be in terms of work. Is there much work available, where you live? Are there prospects? And what has he been good at, or done - was your mum ill a while, and did that affect his chances to study etc? Has he qualifications; does he want to go to uni; has he any work history so far? Was he doing the caring for your mum, and has that impeded him? Does he need some help with grief counselling, maybe, and if so is there a charity that can provide it?
The problem with acting on emotion is you can promise more than you can actually cope with delivering, and then the thing breaks down. Honestly, young men without roots are vulnerable in a range of ways, not least homelessness. I think you all need to talk as a family about how best to support him in starting to build a future, and what you can reasonably do to help with that. It's going to be easier to unite to help him, than it will be each taking a turn until you find it gets too much. And it's not going to help him if he is shuttled around you all in turn, instead of starting to create his own adult life, either.
Finally, I'm so sorry for your own loss. 