I receive telegram of in-game messages while in sleep mode, normal ?

When the bot goes in sleep mode, it is the same as if you were suddenly leaving your computer to go to sleep, and left the browser open.
You are no longer making any actions, but you are still receiving in-game chat messages.

Does that mean I will be shown "online" all the time ?

No. It takes around 15 minutes of inactivity and then the green dot turns dark (offline).
So you will be shown offline to your alliance and buddies.