ATP
Indoor or covered show courts shall have a minimum top height of forty (40) feet (12.19m), except as otherwise approved by ATP.
WTA
At indoor tournaments, the ceiling must be a minimum of forty (40) feet or 12.19 metres in height.
Davis Cup
There shall be a space from the court surface to the ceiling of not less than 9 metres clear (12 metres in the case of World Group Ties). This measurement shall be taken at the net. Applications to use a court with less than 9 metres clear (or 12 metres in the case of the World Group) must reach the ITF as soon as possible after the Draw or the completion of the previous round and at least six weeks (eight weeks in the case of Ties in the World Group) before the date fixed for the commencement of the Tie, whichever date is the earlier.
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ANSWER
As high as the 1st serve that can be, but practically about 25 feet high. and then add a further 8-10 feet for lighting space, etc. So in total a 33 ft height (10m).
ANSWER
There is no mention in the Laws of Badminton, of a minimum height for the ceiling above the court. Nonetheless, a badminton court will not be suitable if the ceiling is likely to be hit on a high serve.
The rim is at 10'-0" and you must take into consideration a player attempting a last second shot from the other end of the court. There will be quite a bit of arc on that attempt.
http://www.sportsknowhow.com/volleyball/dimensions/volleyball-court-dimensions.html
30 feet / Depends on the architect......! No limitations ! Depends on the structure itself!