When do you start the count
 

A1 has control of the ball in the Back Court. A1 throws the ball towards the Front Court. B1 jumps from A1's Front Court and bats the ball back into the Back Court.
1-Do you start a new Back Court count?
If so, then
2-When the ball contacts the floor in the Back Court
3-When a member from team A touches the ball in the Back Court
4-When a member from team A controls the ball in the Back Court

When ball touches BC. There is still TC and ball has BC status.

2.

I don't understand. Are you starting a new count? If so when?
If not, then why?

My mistake. I meant "2." (fixed)

There's a specific case play or interp on this -- and I agree with 2.

I definitely own this one....9.8.1 D (a)
 

9.8 SITUATION D:Team A is in control in its backcourt for seven seconds. A1
throws the ball toward A2 in the frontcourt. B1 jumps from Team A’s: (a) frontcourt; or (b) backcourt and while in the air bats the ball back to A1 in A’s backcourt. Does this give Team A 10 more seconds to get the ball to the frontcourt?
RULING: Yes, in (a), a new count starts because B1 had frontcourt location when
touching the ball thus giving the ball frontcourt location. In (b), the original count
continues as Team A is still in control and the ball has not gone to frontcourt. (4-4-2; 4-3; 4-35-1)

thus, according to 9-8-1...
SECTION 8 TEN SECONDS
A player shall not be, nor may his/her team be, in continuous control of a ball
which is in his/her backcourt for 10 seconds.
PENALTY: The ball is dead when the violation occurs and is awarded to the
opponents for a throw-in from the designated out-of-bounds spot nearest the
violation.

The 10 second count starts when the ball attains backcourt status, regardless of where the ball originated. To do otherwise disadvantages the defense, who have successfully defended their frontcourt from penetration by the offense.
QED

Not A Stand Alone Statement ...
 

Whoa. Not so fast. Hold your horse justacoach. Although your statement (above) works for the original post, it should not be taken as a general statement, that works in all situations, especially if a young official is a little fuzzy on the definition of "backcourt status". For example, on a throw in, even though team control exists, an official doesn't start the ten second count when the ball hits the floor (bounce pass) in the backcourt, in fact, an official doesn't even start his ten second count when the ball touches an offensive player, the official starts his ten second count when the ball is possessed (holding or dribbling) by an offensive player.