Wow, I don't know what Qatari family you visited, but I can't say they all have farms...
Goodness, I can't even say I know one Qatari who owns a horse. You are right about the pools and incredible homes.

Where do the Qataris hangout...I think everywhere. The park across from the corniche and LENOTRE are hot spots for Qataris in the evening. It's mostly just men hanging out outside the restaurant.

They are such wonderful people, I would encourage you to make some Qatari friends.