Everything depending on you is not an part of your personality. I don't find "Everything depends on me" particularly natural unless you're referring to each aspect of a specific event.
Me: Will everything be ready in time for John's party on Saturday night?
You: Yes, I think so. I've arranged the food, the drink and the decorations. I've just got to finalise the DJ and the playlist. It's quite daunting. Everything depends on me!
"I'm just like that" refers to an aspect of your personality or a particular aspect of your behaviour. Tedmc's suggestion in post #3 works better. It means "I'm one of those people who wants to have a say in everything".
