At the end of the movie, the ending is ambiguous (Japanese) or he lives, (American). I posit that the answer is both, that is, he died in Japan, but he lived on in another dimension, the dimension from HMC.
Sho's surgery didn't work, and his heart was going to fail early, and he was going to die young. Hearing of this, his estranged uncle (maybe "Jessica"'s estranged husband?) from a different dimension heard about it, and decided to bring his dying nephew to his dimension, and the house and the flowers, so he'd have a chance to see it before he died. While there, though, Sho found about the falling stars, and made a deal with Calcifer when he fell. Calcifer kept his heart (and him) alive, but Sho's personality changed due to a mix of his heart being gone, and actually having a chance to live.
However, because he didn't know if actual magic existed in his world (oh, yes, there were the Borrowers, but did that mean magic?), so he "died" in Japan, and decided to live the rest of his life in the dimension where he could learn magic, and where his heart was "safe".
Years later, "Howl" makes a name for himself, and meets Sophie, and the events of the movie happen. Now, in the book, Sophie was able to speak life into things, and will things to happen because of magic. In the movie, Sophie does the same thing, unconsciously, and that along with years of being magically healed by Calcifer, his heart was fine, and he lives the rest of his life with Sophie.
- Note, in the movie/book, the uncle's place is somewhere where Howl goes during his magic lessons to get away from Sullivan, also, Howl was already aware of magic, and an apprentice to Sullivan when he went to make the deal. This WMG could still work if he was apprenticed to Sullivan as he was dying, as a dying wish, but he would already have to be learning magic before he traded his heart away.
Sadly, it turns out that someone with experience being virtually abandoned by workaholic (albeit equally emotionally wounded) parents; the traumatic fear of trying to connect with other people out of lonely hope, but only managing to expose both to being hurt in the process; the firsthand knowledge of fantastic, eldritch worlds and life existing secretly alongside our own, rich with joy and togetherness thought unimaginable, if not for the meddling of individual humans, and a resigned familiarity to one's own mortality (often finding himself bedbound on doctors' orders), while undoubtedly sympathetic, probably isn't the best choice to foster raise an equally-traumatized young boy. For anyone.
- And maybe she thinks the Borrowers are disguised agents from the cats to bring her back to marry the new cat prince.
Well... depends on her career, doesn't it?
Sho's mom knew a whole lot more about the "little people" than she let on, even to him. At some point in her childhood, she met and started talking to a few of them, just like Sho did. And finding out that there's this secret population of people - actual thinking, speaking, feeling people - living underfoot, unknown to larger society, tremendously vulnerable to natural predators and the random whims of fate? A thing like that could be pretty formative in a young woman's life.
That one Borrower family in the house that mysteriously disappeared? They didn't die out or just vanish. They left — to partner with Sho's mom. They knew the other families wouldn't understandnote , so they didn't say anything. Together, they set out into the wider world to make a difference.
So imagine this: on the one hand, you're doing this big, important work that's the difference between life and death for people. At the same time, you've not on your own anymore. You've got a son. He's counting on you. A lot of people are counting on you.
But one day you get a call, and all of a sudden you're in Sadistic Choice territory. Either you make time for your son - who could die, remember - or you fly out and you desperately try to buy time for a community of 1,000 people in Chicago to evacuate before the city implodes the derelict building they live in. (Also, just FYI, you haven't spoken English since high school.) People that no one else knows about, could know about. What do you choose?
Maybe you decide... as much as it might tear you apart... you don't get to be selfish in a situation like that.