Overweight baggage policy in Qatar Airways?

Qatar Airways has a page on their website with a handy table that explains all this:

All other flights to and from Argentina, Canada, and the U.S.

• 2 pieces, each not to exceed 32kg (70lb) and a maximum dimension of 158cm (62in) (First and Business Class)
• 2 pieces, each not to exceed 23kg (50lb) and a maximum dimension of 158cm (62in) (Economy Class)

So it appears you can only check in two bags on all classes of travel, the only difference being the weight of each individual bag.

To check in a third bag, you would have to pay the excess baggage fees. These vary based on your flight itinerary.

In your case, for flights originating to/from the US at the time of this answer the charges are $200 per additional item; and $50 if your item exceeds the baggage allowance for your class of travel.

If you buy excess allowance offline (like at the counter or the airline office) an additional surcharge is added.

If you were traveling Economy your total charges would be:

• $50 x 2 = $100 (for each overweight bag)
• $200 (for an extra bag)
• $300 Total

If you were traveling on First or Business, you would only pay $200 for the extra bag as your weight limit is already 32KG.

I don't know of any airline that allows you to substitute a checked-in bag for a hand carry.