In Hereditary, we learn from Annie that her father starved himself to death.
I don't fully understand how that happened.
Did he really have mental health problems, or did the cult/grandmother orchestrate his death as well?
This is something I've been wondering about as well and the theory I've come up with is that the grandmother, Annie's mother, was perhaps feeding her husband, Annie's father, something that was linked to the demonic ritual to perhaps facilitate Paimon's possession of him. Maybe Annie's father realized that his wife was "feeding" him the demon through the food she fed him and he resisted becoming Paimon's host literally starving himself to death and stop whatever ritual Annie's mother was performing.
I thought of this when we find out how Annie's mother was obsessed and insisted on always feeding Charlie when she was a baby. Maybe it was like she was almost "feeding" Paimon to Charlie in some way and we know that whatever she was feeding Charlie worked bc Charlie did become a host of Paimon since she was a baby, maybe even from before birth (perhaps the grandmother fed Annie some ritualistic food while she was pregnant with Charlie). Remember how Charlie never cried as a baby or even when she was born? The grandmother feeding Charlie was definitely something ritualistic and linked to Paimon possessing her otherwise why would the grandmother, who Annie even said was normally very distant and not really that nurturing, be so insistent on feeding Charlie.
In myths, like Persephone and others, you are warned to not eat anything offered to you in the underworld bc the act of eating something somehow binds your soul to hell. I believe I've read about how in certain African witchcraft/juju rituals, victims are offered food by the witch casting a spell on them only for them to become possessed once they eat what is offered to them. There are a lot of other rituals in witchcraft from all around the world where demonic and spiritual possessions are linked to the victim eating a substance that somehow opens the Portal to them becoming possessed or bewitched.
There's also all those scenes of Charlie constantly eating chocolate (which was called the devils temptation) and the event that sets all the horrific events into motion, her death, was started off with her eating a chocolate cake with nuts at the party so you can see the link between food and Paimon's/grandmother's plans being set into motion...
Anyways I think Annie's mom first tried to use her husband as her first target for Paimon's male host. Then Annie's mother likely moved on to her son, Annie's brother, next once she could no longer use her husband as Paimon's host since he killed himself. Then when her son was able to resist Paimon's possession, she started pressuring Annie to get pregnant and give birth so that she would have a new male host only to be thwarted when Annie broke contact while pregnant with Peter and during his childhood, thus Annie's mom never had a chance to "feed" the Paimon to Peter. Along the way, the grandma and her cult likely realized after her husband and son were able to resist being fully possessed by Paimon by taking control back of their mind and body by killing themselves that her next target, Peter, needed to be completely mentally shattered so that he would not be able to resist or regain control of himself once possessed by Paimon.
We don't get a definitive answer to this question. However...
We know that Paimon prefers a male vessel, which could lead us to assume that Annie's mother, Ellen, would have already tried to summon Paimon inside the males of her family, first.
In fact, Annie, while telling her story to the group therapy, says that her brother killed himself when he was 16 years old and delusional, after claiming that Ellen was trying to "put people in him". This is, I think, a direct reference of the fact that Ellen was trying to posses him with Paimon.
So we do know for a fact that Ellen, the grandmother, had indeed tried to possess her own son (who killed himself), which gives us good basis to believe that she would, as easily, have tried to possess her husband, who also turned mad because of it and ended up starving himself to death.