1. Finding extra baggage cost for a QR ticket involving another longhaul airline can be tricky.
Santa Lucia is not a common destination for QR.
https://www.qatarairways.com/en/baggage/excess.html
But I assume that you will be subject to weight concept rather than piece concept. Hence, the price per kg applies. Reservation agents are not well versed on these issues. The problem is that the cost at the airport is higher than if you pre-purchase. Bulk rates are only available on QR. Preferred rates must be done at an office or by calling, and by bundle of 5kg. USD25 seems about right.

2. Bags will be checked through.

3. BA mostly charges for seat reservation, except for expensive ticket fare bucket and OW status members. You might try to login on BA.com with your booking number and see what it says.