I think the objective is to separate the small ring from the chain. Also, it seems the chain goes around the S-shaped part twice. This procedure removes one of those loops, so you'll want to repeat it after you are done:
grab the bit of chain just above where it goes through the S-piece
pull all available slack to that part of the chain, make a loop out of ...
Question: Is a 4x4, 5x5, 6x6 game possible?
Yes! This puzzle can be played with arbitrary size such as in Tatham's version. You can play it in your browser or download a Windows executable from the main page that is full of other puzzles too!
The game is usually referred to as a Sliding puzzle
The most famous example is the 15 Puzzle ($4 \times 4$ version)
In this example, half of all the possible starting positions are solvable. In particular, if we switch the 15 and 14, then the puzzle becomes unsolvable. (see Sam Loyd's unsolvable 15-puzzle).
Higher order examples
Although I have not seen ...