South must be credited with 7 spades.  He has made a diamond and has 2 Aces in dummy so that is 11 tricks.  The first conclusion therefore is that South must be void in clubs if you are to have any chance.  The second conclusion is that South must hold the ♥Q, else West would have led a heart from KQ.  Now things are beginning to shape up.  If you let partner win the ♦Q, South will doubtless try to bring down the ♣K in 3 rounds, reducing West to ♥Kx and ♣K in front of dummy's ♥A7 and ♣Q, with atrump to come. To defeat the impending squeeze, you must overtake the ♦Q with the Ace and switch to a low heart.  Even if South holds ♥ QTx, you are still giving yourself the best chance (he might play the Queen).

It looks like East is now out of clubs so trying for club ruffs in dummy is no good.  You lead a top spade and West follows with the Jack.  On a second top trump, West discards a club.  Now the hand is easy.  Simply play a top club from hand.  If East ruffs if he must lead a heart into your AQ for 2 discards.  If East declines to ruff a club, you exit with a low trump to achieve the same result. 

The bidding marks the missing Kings with West.  The lead indicates that West has at most 5 spades.  A simple line therefore is to win the ♠Q and cash just enough diamonds to exhaust West of that suit and then exit with a spade.  Having cashed his spade tricks, West will have to lead away from one of his Kings to give you a ninth trick.  Is this contract 100% on the play to trick 1? - No not really- it is just possible that West holds a 6 card spade suit and has led his fifth highest to fool you!  If that is the case, remember to congratulate him on his defence.

You must appreciate that the correct play from dummy is the Queen, playing West to have led from Kxx.  There is no point in playing low in dummy as even if you find West with ♠Jxx, East will simply play low and allow you to score the ♠10, but denying you an entry to dummy's hearts.

There are two things which might go wrong - you might have a bad break and a trump loser, or you might find that they take a ruff at trick one before you get to draw trumps. In practice five of the six players with this problem leapt to 7♠, the other one bidding 5♥ and then 7♠ over partner's 6♣ bid.  After your leap to slam, the bidding goes P - P - X  with double asking for an unusual lead.  East led a club and the four heart opener ruffed that for down one.  Sad!

After the event, the discussion turned to his to bid the hand better.  Two strands of thinking developed -

a)  if you can persuade partner to bid spades before you then this hand can go down as dummy and the preempt is on lead and a minor suit ruff is now inconceivable.

b)  if partner is very short in spades, then 7♦ might be easy while in spades you have a trump loser.

Can you make handle either of these options?  A takeout double over 4♥ is a dangerous choice as partner might pass.  The cue bid over 4♥ is read by partner as showing spades and a minor, so it might work but here it got a club response.  The final thought is bidding 4N which, because it bypasses spades, shows the minors.  If over this partner preferred diamonds, there would be a very good case for bidding 7♦ .  When partner bids clubs on today's hand, you could always try a cue bid of 5♥ and here partner will cue bid 5♠. Just what you want! You raise to 7♠ and partner scratches his/her head.  With any luck they can't find anything to do and they pass.

This is all a bit fanciful, but I bet the players concerned won't bid 7♠ on the first round next time they have this hand!

You have 12 tricks on top.  The trump lead deprives you of a heart ruff so you will have to look elsewhere for your extra trick.  Whilst the club suit offers the best chances (you will ultimately take a club finesse if nothing exciting has happened), you should give yourself an extra chance (albeit slim). Draw trumps, cross to dummy with a heart and play Ace and another diamond, ruffing.  If a top honour falls you can go back to dummy with a club to lead another diamond.  If either defender started life with ♦ KQx, you have your thirteenth trick.

This one should be easy.  On the bidding it is very unlikely that East holds the ♥K  You should appreciate that partner is trying to kill dummy's entry to the club suit and you must play your ten of hearts on this trick.  You know that declarer has 2 clubs and a 4 card heart suit.  Even if partner does by some miracle hold the ♥K, then you will still collect 2 hearts tricks to beat the contract.

You have a heart and a club to lose so it appears that you need to avoid a diamond loser.  The diamond finesse is a 50% chance - better than trying to drop a singleton King.  However, the best chance is to enter dummy with the ♦A and take a spade finesse.  This represents the same 50% chance as the diamond finesse but you have the added bonus that you might drop the ♦K, thus increasing the overall chance of success.

One line might be to draw trumps and try to get the diamond suit right - but I wouldn't fancy the diamond guess if West follows with small cards.  If trumps are 2-2 then you would have 2 dummy entries to take a double diamond finesse - a 75% shot.  You could also get your entries to dummy by ruffing a heart and later using the ♠Q as a second entry. Either line might fail if diamonds are 4-1 and the defence can manage to get a ruff.  An alternative play is to just cash 2 more hearts, throwing diamonds from dummy, and then play Ace and another diamond.  This line works whenever diamonds are 3-2, and also in many cases when they are 4-1 and you can engineer 2 ruffs in dummy.  This is the best percentage line.