I am working the last 3 weeks of March in Manila with little free time. Could I have an enjoyable short weekend (Friday night to Sunday night) in Bangkok, getting to and from BKK at reasonable cost and seeing a number of major cultural sites in BKK? If so, what airline should I take, what should I see, and where should I stay?

If it were me, I wouldn't think it was very reasonable. The flight from Manila to Bangkok is three+ hours, so you'll arrive very late on Friday and have to leave Sunday afternoon.

That just leaves one full day of sightseeing, which isn't really enough. If you did do it, you'd be exhausted. Keep in mind that end of March / beginning of April is the hottest time of year in Bangkok.

I would tend to agree with the above poster that a weekend from Manila would be tough, having looked at flight options. It appears that the return flights on Sunday are rather early, either a non-stop at 2 pm or the latest seems to be a 4:25 pm with a connection through Hong Kong that would not arrive back in Manila until 11:30 pm. Even with a 4:25 pm departure, you would have to leave for the airport by 2:30 pm or 3 at the very latest, which would eat into your Sunday considerably. A “tour” of Bangkok in 1.5 days would be tough, IMO, to really see and enjoy it; possible but quite rushed (and to spend the airfare one maybe not worth it). Hong Kong or Singapore would be better choices perhaps, they are more compact cities with shorter flights and more flight options.

If you could take the Friday or the Monday off, then Bangkok would work well, IMO. Still a lot to see and do, but much better timing.

You might want to look at flight options, 'cause I used Expedia, which is not always the most accurate, so perhaps there are other flights. Cathay, Phillipine Air and Thai seemed to be the principle carriers for the flights.

There are some great beaches in the Philippines as well.

maybe thai air or phillipines air...??

hong kong would be a good alternative too...

As would Hanoi or HCMC.

Try the low cost-airline "Cebu Pacific" cebupacificair.com. They have regular flights to BKK from MLA that leave at 10:00 pm and arrive in BKK at 1:30 am. Return flight is at 1:30 am and you would be back in Manila at 5:00am Monday. RT ticket would be around US$125++.
You may want to checkout the "Swingaround" promo of Philippine Airlines - www.philippineairlines.com - they have a BKK package tour that starts at US@254, with a 3d/2nite hotel accommodations.