ExtraBold> But I don't want to buy a book on 14 Nd7 in the Catalan variation
First buy a book that teaches opening principles and covers all the major openings, like Ward's Improve Your Opening Play or Fine's Ideas Behind the Chess Openings. This will help you decide which openings you like and get your started playing them. They will also teach you opening principles that will help you no matter what opening you play.
Your opening preparation should very rarely be going 14 moves deep. That would be a really, really silly waste of your time. Focus on learning ideas.
ExtraBold> But how do I know what my style is, or what suits it?
The above books will help. Also, try playing ten or fifteen blitz games in openings you're considering and see how you like the resulting middlegames.
ExtraBold> 2. When choosing a repertoire, how do you systematically get the right coverage of likely enemy moves so there aren't big holes?
Once you choose an opening, buy an introductory book on it, eg "Starting Out: the xxxx".
ExtraBold> 6. Is there a mnemonic system for remembering opening lines and ideas? Surely there must be. Or are they all trade secrets?
The trick is understanding opening principles and the goals of your opening system. Then not so much memorization is required. A student of mine asked me how I memorized a 20-move game after seeing it only once. Easy, I told them. 16 moves were common sense, so I only had to memorize the four unexpected moves!
It's better to understand an opening four moves deep than to memorize it eight moves deep--and I'm saying that from experience. Alas, some memorization is required (when there are two good moves). Chessbase, Chess Opening Wizard (aka Bookup), Chess Position Trainer, and index cards all work well for that.
At the moment I am playing Ruy Lopez, 2 Nf6 to the sicilian, and some sort of Tarrasch in response to d4. Someone opened c4 the other day, and I think some sort of reverse sicilian is in progress. Nobody has opened 1Nf6 yet, but I guess the Tarrasch would work if they did. It's OK I guess, but:
1. I want to wean myself off Game explorer, without too much work.
2. When choosing a repertoire, how do you systematically get the right coverage of likely enemy moves so there aren't big holes?
3. Choose openings that suit your style. But how do I know what my style is, or what suits it? I have played over 100 online games here, and rating in the high 1800s - most games seem to be won or lost with basic tactics and combinations, or blunders. Does this mean I like tactical games? I like them not too tactical for me, but too tactical for my opponent!
4. Is there an opening guide with a handful of moves, reasons for the moves, and strategic ideas going into the middlegame? For any given opening. Or do I buy a book? But I don't want to buy a book on 14 Nd7 in the Catalan variation of the bishop's Indian attack of the London system before I've even decided if that's what I want to do.
5. I only took this up because my son was interested, but he has since moved on from chess, to Little Big Planet on the PS3. All his games begin 4 knights, 2 bishops, boredom. I've suggested different moves might make the game more interesting for him, but I don't know what exactly to suggest. It needs to be something with ideas for the middlegame in the package. Or is this futile, bearing in mind that everybody he plays will hang pieces sooner or later anyway. He doesn't play me any more because I don't always begin with 2 knights and a bishop.
6. Is there a mnemonic system for remembering opening lines and ideas? Surely there must be. Or are they all trade secrets?
7. Yes I have practised tactics quite a bit, and some endgames, although the harder endgame stuff doesn't make a lot of sense to me.