I'm quite new to the game, do the cards in my collection have a direct bearing on what cards you choose from in the Arena? Is it worth crafting a good card in the hope that this will then be offered in the arena? Or does the arena contain cards that I don't even own yet?
No. Your card collection has absolutely no correlation to the cards you can pick in the Arena. In fact, you can end up with Arena decks that are illegal (for example, because they contain more than two copies of the same card), depending upon what random cards are presented to you and the cards you choose.
The random cards presented to you will be from all the non-promotional cards available in the game, regardless of whether you currently have a copy of that card or not. This means that you can expect your opponents to have any possible card in the game as well, which makes for a bit of a different strategy / play style than regular match play.