What is the rule on objects at tournaments? I know there needs to six of them, but is there a regulation as to how many light and heavy objects there can be on the board? Does it have to be three light and three heavy objects? Can each player furnish their own three objects?

From the Icons rule book (page 11):

STEP 4: PLACE OBJECTS
A HeroClix starter set includes six object tokens—round cardboard tokens that represent items characters might find on a battlefield and make use of in battle. Tokens with a yellow border represent light objects, and tokens with a red border represent heavy objects. Several 3-D objects, three-dimensional sculptures that can also represent items on the battlefield, are available separately. All 3-D objects are classified as either light or heavy, and some have special rules associated with them. After all players have identified their starting areas,
each player places any three object tokens facedown in a common pile. After all players have added three object tokens to the pile, mix up the pile. One at a time, starting with the first player and moving around the table clockwise, each player takes a token from the pile, looks at it, and places it faceup on the battle map. After choosing and looking at the token, but before placing it on the battle map, a player who has 3-D objects in his or her collection may choose to replace the object token that was selected from the pile with a 3-D object; remove the object token from the game, and place the 3-D object on the battle map. Objects must be placed on clear terrain (see “Clear Terrain,” p. 23) and cannot be placed in any
player’s starting area. Continue in a clockwise direction until all tokens are placed on the battle map.

Also your opponent can't prevent you from using 3d objects if they don't have any and you don't have to let them use the ones you bring.

And now with the special objects you can use them as well, even if your opponent doesn't have any. The only restrictions are they are unique, so you can't use 2 fuel tanks, you must swap a light for a light or heavy for a heavy and the blue ones can be swapped for any object but you can only have 1 blue object for every 300 points. So 200 points, no blue object and less than 600 points, only 1. Per player. And they can't be placed within 5 squares of any starting area.

When the Danger Room comes out, the 3d objects will be errated to have the gear symbol which means they will have the same rules as the special objects.