As for the Knight, I'm afraid using N for its notation makes it all too tempting to call it the Nag ...
Why is a castle called a rook?

The Persian word is Rukh. There are two alternative names for rook in Russia; Typa which means tower I think; and ladya, meaning a sailing boat of Northern cultures. Also, in Bengal the Persian word for Rukh translated to Roka meaning boat. I'm assuming this would be much later than the 'castle' use of the piece as the game would have had to spread to Persia before it became the Rukh. In Indian it was the ratha (chariot) as part of the chaturanga game. In Persia it became the rukh (still in chariot form). As the Arabic caliphates conquered the Persian Empire it brough chess into Southern Europe where it modified into todays game. As Russia also used the tower as a rook I think (from what I've read) that the boats that developed in Bengal and Russia were not from the same 'game'. Right up until its development in Europe, there were loads of forms of chess. I guess thats why Europe is the most important, from the 9th century to the 15th Europe developed it into the universal form that we have today (probably because of the ease of transport throughout Europe and therefore better communication).
The rook would be in the shape of the chariot because it was originally the chariot in India. Probably you're thinking of the bishop being the shape of the elephant as this was again the case in India. The cannon actually comes from Chinese chess, the proper name of which is xiangqi. The modern form of xiangqi had developed by the 12th century, a lot earlier than chess (15th century) but a lot later than the initial 'design' of the rook (i.e. the chariot that has been found as early as the 6th century). Its really not know whether xiangqi developed from Chaturanga or whether it was its own game.

Before one could be a knight, one must first learn to ride a horse. Though knights were sometimes knocked off, they were mainly cavalry

The knight was a cavalry piece. When it went to Persia it was simplified. Religion stated that sculptures of human form were not allowed so it turned into a very simplified 'horse' shape but it still represented cavalry. When chess came into Europe, the cavalry unit of the time was the knight so as the horse shape gradually regained it's detail, it became the knight.

Really interesting thread!
Do you know anything about the origin of the term used for bishop?
In italian language the chess pieces are called quite differently respect to english:
- rook is the "Torre" ( = tower)
- knight is the "Cavallo" ( = horse)
- bishop is the "Alfiere" ( = "standard bearer")
For the first two pieces I found the italian terms more appropriate (the standard pieces really seems a tower and an horse) , for the third it is funny how different is the meaning... bishop (in italian "vescovo" a clerical hierarchy related term), "Alfiere" has a completely different meaning.
In any case my dictionary for the "Torre" word, related to chess, propose both "rook" and "castle"
Like the previous poster says: Al Fil is Arabic for Elephant, and is the name of the Bishop replacement in Shatranj, the ancient Arabic form of Chess from which the FIDE game evolved. (The Elephant jumped only two steps diagonally there.) When the Italians learned about the game they mistook Rukh for Rocca, and Al Fil for Alfiere.
Note that in no other language I know of other than English this piece is named after a clregy man:
German: Laufer = runner
Dutch: Loper (= runner) or Raadsheer (= cousellor)
French: Fou (= jester, fool)
Spanish: Alfil (= Arabic for Elephant)
Italian: Alfiere (= Standard Bearer)
As to the term 'castling': in general this means building a fortress to protect the King. That in Chess it involves the Rook and that there is a special move to make it easier, is just coincidental. In Shogi (Japanese Chess) constructing a viable castle that protects the King in a corner is the main purpose of the opening, and it never involves the Rook (which is used in attack). There is no special move for it, so the King needs to walk the distance towards a corner on its own power; the piece sitting in the corner (a Lance, which only moves straight ahead) usually stays there, to protect the castle's edge.

I know this has nothing to do with how it got its name but it is interesting that the chess rook is well described here. Like a rocket which can fly from one side of the board to another. I could visualize how the rook can move like a rocket, going in a straight line from one side of the board to another.
Have you considered that the decision to assign that moniker to the rocket was inspired by the rook in Chess. Just curious. Too drunk to Google the reason why the rocket was named "rook".

This is interesting because in the chess variants used in Thailand, Cambodia, and Indonesia called Makruk, Oua Chatrang, the room is a also a boat.

OED says the term originated with dragoons in the English army as a slang term for new recruit.

It could have ended here with a footnote to H.J.R. Murray, but people must voice their opinions.
======
Techno_77 wrote: "Question followed up by, why is the knight a horse?"
======
Regarding my answer about the rook and the blackbird, I was answering only the part about the blackbird (the merla, the crenellations) which one might realize is the Phoenix from the ashes... which is related to the actual full "turret" structure of the "tower"...
Without intentions to hurt anyone's feelings, the whole game of chess symbolizes our entire magnetic and electric experience... I do not want to step on anyone's beliefs... I was only answering the part about the bird...
To answer a question about a "whole" piece like the whole rook or the whole knight... that, I am certain would ruffle many feathers... so, I will just interject a toe into the waters and see if anyone wants to kick me or if people want to really know...
Q: what if i told you that the "whole" "rook" piece and the "whole" "knight" piece were the "offspring" of the bishop?
Thank you, again...

The Unabridged OED (about 600,000 words) is currently being revised (projected completion 2037) but the additions are slowly being incorporated into the online edition. There are no plans to publish a third edition as it is estimated that it will be approximately double the 20-volume size of the (current) second edition.
It is possible to subscribe to the online edition ($100/year), but many libraries have a subscription and library-card holders can access their library's subscription online.
I worked for years at the University of California, Berkeley and had access to their libraries' copies of the printed editions, but the convenience of online access can't be beat.
1913 OED Vol. 1 (a-b) 1282 pages is located here...
https://archive.org/details/in.ernet.dli.2015.99992
the rest of the words are found on archive.org and other places... various editions...
Thank you, again

why is a horsey called a knight
Its called an hoss in our house, plural of hosses.
Rook is short for Rookery, which is what those towers on a castle are.