the bus was crowded when i boarded the bus
the horse and the carriage is waiting outside the house
I would suggest- " The bus was crowded when I got on."
"The horse and carriage are waiting ........."
"The horse and carriage is waiting........"
I think you could use either because "horse and carriage" can be regarded as one vehicle or as a horse and a carriage separately.
Personally I would tend towards the singular - is.
Notice we don't use the carriage in this context.