The website is confusing, but no. The child has to have "a ticket," but that doesn't mean they have to have a seat. It just means that you have to pay for the lap infant for int'l travel, whereas a lap infant travels for free on domestic flights. I believe the cost is 10% of the ticketed cabin's full fare, but I'm not positive about this, having never done it.

As for the discrepancies between prices for the same flight... that's just how codeshares are sometimes, and I have no idea why. I believe the non-operating carrier basically purchases some number of seats in bulk from the operating carrier, and they can then sell them for whatever they like. Of course, I'm just guessing here.