Greed’s death can indeed be considered essentially committing suicide by convincing Ed to kill him. This is basically confirmed by Martel in episode 39. She thinks that Greed wanted Ed to kill him, as that was better than being left with nothing.
The reason he insisted on challenging Ed was, I believe, so Ed could find the strength to kill. He could’ve revealed that Alphonse had already been freed but he let Ed think otherwise and didn’t correct his assumption on him killing Dante.
He wanted revenge on Dante and the Homunculi for what they did to him. Greed had accepted his life was over when he stepped into Dante’s house and he didn’t want to be trapped again. He couldn’t bear the idea of working for her either, what he most valued was his freedom. Challenging Ed and telling him the Homunculi’s weakness was the only thing he could do to give his death any meaning.
The Reflections OVA actually suggests that the reason he wanted to know how to replicate Al’s soul armour for himself was so he could still have immortality without needing to rely on Dante for red stones, allowing him to be free of her. All his actions could be framed in this light, allying himself with powerful alchemists and chimeras (and chimera alchemists) as well as his interest in getting Wrath on his side were to gather enough strength to be free from Dante’s control.That all failed with Envy winning Wrath over and his chimeras companions being killed.
Zetalial 