cysyuen before any answer can be given we need to know how many reservations (PNRs) do you have? which flights are on which reservation? what cabin class are your flights?
As a reward flight LHR-BRU wil not be considered by QR since your baggage is not allowed to pass the same airport (LHR) twice and also because BA will not interline your bags. Officially there is a security rule that your bags must travel with you the entire journey so checking in at LHR before flying to BRU to start your QR journey is not permitted. But many airlines ignore that rule. The rule would not apply if BRU-LHR was also an award ticket!
QR will interline with OW partner UL even on separate reservations.
Yes the MSC rule will apply on BRU-LHR-DOH-CMB-HKG if you interline your bags. But do be aware that you baggge allowance is goverened by the which airline is shown on your reservation as the marketing carrier for the DOH-CMB flight. If it is a QR flight number then QR allowances apply, If UL then UL allowances apply. The MSC is the marketing airline that first crosses an IATA area boundary!
Check-in starts three hours before departure. Sometimes the higher cabin classes are allowed to make a bag drop earlier provided they already have printed boarding passes (OLCI).