You cant rent a beach for a party esp if you have alcohol involved.

About hotel (cheap) You could try grand hotel in musheirib or Sofitel which is mercure hotel now.

Zig zag only people who stay can book the hall /party area for u .

If you know anyone who lives in a compound with a public Club house would be the best like they have in Matar kadeem (Old airport area) ,you dont have to pay for the club houses.

My personal advise from prior experience in party organizing , Private parties to be hosted in Hotels or Club houses are safer as u know we have too many rules in Qatar to follow once you wana throw a party out.

Best of luck ,make the most of the nite !!Cheers