What chess books should a Class A player read to reach Expert Class? What are some book recommendations, tips, and ideas for a Class A to reach Expert Class? What knowledge do most Class A lack compare to Experts?
I'll start with a few tips and ideas.  None of them are original, but all of them have some value and bear repeating.

(1)  If you have the opportunity to play in tournaments where there are stronger players, this is the best way to improve.  The hard lessons you learn from your tournament games are ones you remember best. Moreover, every mistake you make in a tournament game is showing you a gap (a poker player might say a "leak") in your play. It is no coincidence that all really strong players played hundreds (if not thousands) of serious games before they got really good.   

(2)  Learn to sit on your hands.  Not literally, of course, although you could try it if you are really impulsive.  Many players make mistakes they could avoid if they just slowed down a bit, sat on their hands, and thought a little instead of moving quickly.  When I started slowing down and making myself take a little time and think about every move, even the obvious ones, I made a big step forward.

(3)  Learn to fight hard through the entire game.  Don't relax until the game is over.  When you have a won position or an almost-won position, this is when your opponent is often at his most desperate and dangerous, and you should be very careful.  Chess is a game where you can spoil everything with a moment's inattention.  Learning to win won positions in an absolutely relentless fashion without giving your opponent the slightest chance is a very important skill.  Learning to create as many practical difficulties as possible for your opponent when you have a bad or lost position is also very important.  Never stop fighting until the game is over.

(4)  Do lots of puzzles.  Tactics puzzles, positional puzzles, endgame puzzles, it doesn't matter much which. They will develop your ability to solve problems over the board.  People often read through game collections, middlegame manuals, and the like on autopilot without doing a lot of thinking for themselves, and then expect to improve.  But abstract ideas are practically useless in chess. Knowing that the bishop pair is usually an advantage is nice, but does you no good at all unless you have some concrete idea about how to exploit this advantage, and can apply this idea in practice.  You need concrete understanding, and to get it you must be thinking and calculating for yourself all the time.  The more you think for yourself, the better you get at it.  Puzzles are one way of forcing you to think for yourself.  Solitaire chess is another.  I am a big fan of solitaire chess (any at least lightly annotated game collection will do), as it presents you with problems to solve which cover all aspects of the game.

(5)  My last idea is rather murky, but nonetheless very real.  The single characteristic shared by all strong chess players I have known is that they have a strong desire to win: they are highly invested psychologically in the outcome of every game.  They really hate to lose, and they really love to win.  As a result, when they sit down at the board they are highly motivated, they have a lot of fighting spirit, and they are hard to kill.  Anything you can do to increase your motivation (find a rival, set a serious goal, play on a chess team, etc.) is a plus.  In my experience, people who are not highly motivated usually don't get very strong, no matter how talented.

I am a little hesitant to recommend specific books: different people react well to different types of material, and honestly I am not sure what "Class A" means.  There is quite a big difference between an 1800 elo player and a 1999 elo player.

However, I would primarily recommend puzzle books for anyone below about 2100 elo.  Any one of the modern "1001 tactical problems" type books would be good (e.g., Zenon Franco's "The Giant Chess Puzzle Book," Richard Palliser's "The Complete Chess Workout," Igor Sukhin's "Chess Gems," etc.).  An endgame puzzle book rather than an endgame manual (e.g., John Hall's "Endgame Challenge," Bernd Rosen's "Chess Endgame Training," or the like).  Maybe a book with some positional/strategical puzzles as well (something like Johan Hellsten's "Mastering Chess Strategy" would be good, there are probably others).  

Maybe add an annotated game collection from one of your favorite players (like Tal or Fischer or Karpov or Kasparov or Anand), or an instructive game collection (like John Nunn's "Understanding Chess Move by Move," or "The Mammoth Book of the World's Greatest Chess Games").  Actually, it doesn't matter so much which book you choose as how you use it.  If you really grapple with the material and try to master it, if you constantly set yourself problems and try to solve them, any decent book can take you a step forward.  

I am not going to address openings, as most players below 2200 elo study openings way more than they should.  Just find some solid openings that you like and stick with them, so that you build up some experience in the lines that you play. If you have a problem with a particular line, look up the position in the database and find out what stronger players are playing in that position.

But mostly, a serious approach, motivating yourself to fight hard through the entire game, and a lot of tournament games are the main things.