Does the son's inability to house his half-brother, after the full brother reneged on his commitments, justify the father's displeasure and anger towards him, knowing that the documents obtained for residency were forged?

You must strive to seek your father's satisfaction, for obedience to him is obedience to Allah. Explain to him that you have exerted your effort with your siblings, and that you are not legally obligated to provide housing and financial support for them; rather, they should rely on themselves. If he is convinced, then praise be to Allah. Otherwise, you are not obligated to do what is not legally binding upon you.