Can you promote a pawn to a queen if your queen is still on the board?

Yes. The Laws say the choice of piece is not restricted to pieces that have been captured earlier, which settles it in one sentence. Your original queen has nothing to do with it, and neither has the opponent's. Every pawn that reaches the eighth rank picks its own piece, and it can pick a queen every time.

What do you use as a second queen over the board?

Ask the arbiter, or borrow the queen from a spare set on the next table. You may stop the clock to sort it out. Do not turn a rook upside down and hope: an arbiter will treat an inverted rook as a rook, which is a five-point mistake, and in a scramble your opponent may honestly not know what the piece is.

How many rooks or knights can you have?

Ten of each, in theory: the two you start with plus eight promoted pawns. Ten knights has been done in composed problems and never in a real game. Underpromotion to a knight does happen in practice, though, usually because the knight arrives with check and the queen does not.

How many queens can you have in chess?

Nine, and the arithmetic is quick: one queen at the start, eight pawns, and each pawn may promote to a queen. Nobody has ever needed nine. Two is the number you will actually meet, and it tends to look like this: a queen already sitting on c7, a pawn one step from b8, and 1.b8=Q ending the argument.

White has a queen on c7 already. The pawn on b7 makes a second one and the game ends at once: 1.b8=Q is mate, because the new queen covers g8 and Black's own pawns take h7 and g7.

Where the myth comes from

Plenty of club players believe you can only promote to a piece that has already been captured, so a pawn cannot become a queen while your first queen is alive. It is not true and it never was. The Laws state that the choice is not restricted to pieces previously captured, which is about as blunt as they get. Promote to whatever you like: queen, rook, bishop or knight, of your own colour, on the square the pawn arrives at.

The confusion is physical rather than legal. The box has one queen per side in it, so when a second one is needed there is nothing to put down, and people assume the rules must forbid what the equipment cannot do.

What to put on the square

Over the board, stop the clock, catch the arbiter’s eye and ask for a queen. There are usually spare sets in the room. If you are playing casually, borrow the queen from a set on the next table.

What you should not do is invert a rook. An upside-down rook is a rook as far as an arbiter is concerned, and in a time scramble your opponent may not read it as anything else. There is a whole page on promoting without a spare queen, because it comes up more often than you would think in junior events.

Online none of this exists. The board hands you a queen and asks no questions.

Two queens, and the trap that comes with them

The second queen usually ends the game inside three moves. It also doubles your chances of stalemate, which is the one way to throw the win away: two queens take away so many squares that a lone enemy king can suddenly have none at all while not being in check.

The habit is the same one that saves the king and queen mate: before every quiet move, count the enemy king’s legal squares. If it is zero and he is not in check, find another move. And if the position is already mate-in-one with the pieces you have, take it rather than promoting for the sake of it. A new queen you never use is a new way to draw.