It's not clear whether they rejected his healthcare eligibility or just his disability claim, but he might try going to a Vet Center for immediate help, if he's reluctant to go to a VA hospital. The VA has forced the Vet Centers to be more clinical, but they're still more laid-back and approachable than the hospitals. They also can help with referrals to other resources in the community to help address his immediate needs. And other vets there--individually and in support groups--can be an invaluable part of a support system.
If he doesn't have a good claims rep from one of the veterans service orgs, he should get one. Both for upgrading his discharge and in appealing his disability claim. The rep will handle all of the communications with the VA and take a lot of that stress off the vet. Some in the local vet community will have a good sense of who the good and not so good reps are.
For a discharge upgrade, supporting statements from mental health professionals can make a big difference. If he isn't getting that support elsewhere, Vet Center psychologists and counselors can help provide that as well as ongoing counseling.
If he has a legitimate disability claim that isn't documented in service records, another way to document it is to get affidavits from battle buddies who witnessed the event. Service org claims reps usually have a lot of experience tracking down those vets, obtaining their testimonies, and putting together disability claims and appeals.
Pursuing eligibility and disability claim issues can take time, and the first concerns obviously are his suicidal ideation and his immediate daily living needs. NCHV is a good place to start.
You are also now an important part of his support system. A caring, compassionate caregiver who really takes an interest and follows through is worth her weight in gold.