He sounds rude and annoying, but I also suspect there is a lot more to his story than the side you've given here.
He's obviously had a very difficult childhood, I wonder whether he was rejected by a parent or/and step-parent and then you, ten years younger, was the golden child. He probably had other struggles in life that you don't know about. Anyway, that doesn't excuse his rudeness, but may give an insight into his anger.
You only communicated through one method, you stopped using that method without telling him. You struggled with a password but you could have reset that, or sent a message through your mum. Because of this, when your mother hurt herself he didn't hear about it presumably until afterwards, which must have upset him.
Do you want to have a relationship with him? If you don't, then he's set things in place for you to not need to contact him again. Being related to someone doesn't put you under any obligation to be in touch with them. You have the opportunity now to drop him completely.
You say you will still send him cards and messages though so perhaps you do want some level of relationship? In which case, you should send out what you want to, but must expect nothing in return, otherwise I suspect you will be disappointed. Only communicate when you want to, never out of obligation.
I think it may be a good idea to let him communicate directly with your mother if that's possible, never through you. It can be difficult for a sibling when another sibling becomes the "gatekeeper" of an elderly parent's communication.