We play a card game once a week which is a variation on whist/trumps,
nomination whist using a standard deck of cards without jokers.

The game has a total of 20 rounds.  During one round the object is not
to win any of the seven available tricks.  The penalty for gaining a
trick is 5 points deduction per trick taken.  However, if a person
chooses to call three or more ticks and gets the correct number of
tricks called, points (10 points) are gained instead.  At the start of
each round the top card of the pack is turned which determines the
trump suit for the round.

The person with the lead, on this occasion, Andy, called three tricks.
 Trumps were declared as hearts with the 7 of hearts turned.  His hand
was:

10 diamonds
2, 3, King, Queen of clubs
Ace, King 3 hearts (trumps)

There were six players in total, each having seven cards.  (Each group
of six cards is one trick ie. one card played by each player.)  The
person on the left of the lead player has to play their card next and
so on until everyone has played one card each.  The winner of that
trick takes the played cards and claims it as a trick.  He then has to
play the next lead card and play continues as described above.

Andy was the only person to call three or more tricks.  It should be
noted that the penalty for not getting exactly three tricks on this
hand is a loss of many more points -approximately 10 points per trick
taken.  It has therefore been known for other players to deliberately
prevent the calling player to get their call correct by either forcing
them to win one more or lose one more trick than they have called. 
Ie. over-trumping to take a trick away.  It should also be noted that
if you have a card in the suit that is laid by the lead player you
have to play it.  If you do not have the suit you can put any other
suit on the trick.  Eg. Diamonds are trumps; If a club is played and
you do not have a club you can either play a card of another suit to
reduce your chances of winning with that card on a future occasion if
you do not wish to win any tricks.  Similarly you can ?trump in? with
a diamond and take the trick if you do need the trick.

Having the lead meant that Andy could lead any card he wished to. 
Winning a trick means that you have to lead the next card.

I need to know in which order Andy should have played the cards in
order for him to win exactly three tricks.  I am sure it is possible
to work out statistically which is the safest order to play them in
but I need a GAR to work it out for me!

Please let me know if you need anything clarified as having read this
through it does appear to be a little confusing!  Playing the game is
easier than writing down how it is played!

You should probably bump it up to $100.

As an avid card player (yet never having played this particular game),
I have some input.
Andy has 2 sure winners (A,K hearts) and some very low cards (2,3
clubs & 3 hearts) that his task is not a hard one.  With 2 sure
winners, he only needs to win 1 more trick and avoid taking extras by
tossing his low cards before other people are out of those suits.

If I were leading, I would start with the 10 diamonds.  That is the
most iffy card he has (could win, but will probably lose).  If the 10
wins, then leading the 2 clubs is an obvious move.  From there on, it
is simply a matter of tossing off the highest card possible that
doesn't win a trick... and taking the last 2 tricks with the A,K
hearts.
If the 10 loses (the more likely outcome since there are 4 diamonds
that beat it and the possibility that 1 of the 6 players would trump
it), then Andy should try desperately to take a trick by trumping with
the 3 of hearts or winning with a high club.  Then after successfully
taking a trick with 1 of those 3 cards, play to lose all the tricks as
described above.

Statistically, I'm not sure any conclusion can be made because it is
unknown how other players would play even if we know what cards they
have.  Some players would probably be more aggressive in trying to
make Andy take fewer than 3 tricks while others would give up hope of
that very soon and try to bag him with extra tricks (even given the
same cards)... people are very hard to predict in cards unless you
already know their playing style.

Stastically this cannot be done with certainty as you cannot gaurantee
all forced moves.

The way the second player plays the card would affect the way the
third plays etc.  If the second player, plays to win the first trick
that does not stop the third player if they had both higher and lower
in the same suit, from letting the second player take the trick or
winning it themselves.

Unfortunately this one has the 'human element' on it, so I do not
think it can be done.