1. "try on" is used in situations where clothes or anything to put on the body is involved.
"try out" is generally used for like situations that someone is trying to do for the first time.
So no, they are not the same.
2. Examples of "try out"
- "I am going to try out for the team."
- "I am trying out the ingredient to see how it tastes."
- "Will you try it out before making your decision?"
- "Can you try out this lock for me?"
Hope this helps.
3. Yes, "try (one's) hand" always means doing something for the first time.
No, "take up something" does not always = try hand.
It can mean the same thing in certain situations, but it can be used differently from "try hand"
Example: - "I will take up the leadership of this group."
Here, it could mean, that the person already has experience leading people and decides to take up the role of leader. So he/she is not doing it for the first time.
It's all about context. That's the funny thing with English.
PS- "try out" and "try (one's) hand" can mean the same thing. Again, it is all about context.
Hope I help.:)