Chess Online

I'm a little surprised this one only has a 9% success rate

Again, FEN info is needed, in order to know any possibly castle and en-passant captures.
A chess position is not complete without all that info. That's why FEN standard exists!

Being able to castle is not something that is foreseeable here.
Anyone who does not like to use the analyse button is screwed over.

In a real game you know all the encastlements that are possible, and the en-passant capture possibility (if any). So, all that info must be known in a puzzle, to consider it a complete position. Anything else is just a waste of time... is just NOT chess.... and we are here to practice chess.

You don't know to click on the king to see that. In a real game you'd know if it's a possibility, in a real game I always think about if the opponent might castle until they can't.

This claim that clicking on the king makes it all okay presupposes that I'll see a reason to click on the king.

So, you're saying people should change how they play to accommodate idiotic and ill thought through chess puzzles, rather than thinking through the said chess puzzles?

Touch move being something that one needs to practise, and can't if we do things like you'd want us to (and yes, I am rolling my eyes at you).

I think there are a few separated but correlated issues here.

1. the user should be able (obviously) to know if castling is valid or not. He can do so because he can click on the pieces to see it

2. the user has to know about this possibility, this might be the only real problem I can see here, although I would think that most users would get this after using the site for a little bit

3. the composer of the puzzle might not want to make it too obvious that the solution involves castling, this doesn't have to mean he wants to mislead the user into thinking castling is not a possible...

4. the puzzle should be like a situation of a real game:

4a. the important think is: in a real game it is easy to know if you can castle, the same is true here

4b. the only differences are aesthetic, in a real game you know you can castle by remembering the past moves, here you know it because you can click on the king; in a real game you cannot touch the pieces while here you can. Does it really matter? I think people are over reacting  

You need to know to click on the king.
And you're presuming people don't think before clicking.

Not if they're presuming fen protocols, which would be both justified and reasonable.

And you can't write that it's a problem and then say that it's not a problem.

It either is or it isn't.

So you saying it's not a problem is fine, but not if you're going to say that people can just click. That presupposes that they'd think to click and look.