I might be confused how EventHubs supposed to be used or need guidance on how to reliably process events posted into Eventhub. I export Azure ActivityLog to Eventhub and currently just using console application to read those messages. What I don't understand is what I'm supposed to do with events which I already read and processed. Say I want to write content of all messages into Storage account AppendLog. For this I need to delete messages which I already processed (like it would be done if it will be message queue), how do I do that with eventhub?
You cannot delete them. From the docs:
Event Hubs retains data for a configured retention time that applies across all partitions in the event hub. Events expire on a time basis; you cannot explicitly delete them.
Back to your question:
Say I want to write content of all messages into Storage account AppendLog. For this I need to delete messages which I already processed
I am not sure why you need this though. You can keep a pointer to the last read message so you are able to process only new messages. Why should you need to delete the older ones? You can read about offsets and ceckpointing here.
What technique are you using for reading the messages?