Can't comment about the WHY seat, however you should be able to take your dad into the lounge at SIN.

BMI flights depart from Terminal 1 so you will need to clear through immigration in terminal 4 and then transfer yourself over to terminal 1. I doubt that your baggage will be checked through to your final destination.

5:30 can be a busy time at LHR so you may need the hour and a half!