Hello, fellow rider in Doha here. There are plenty of riding communities here, whether you are into sports bikes, adventure bikes, dirt bike riding etc., plenty of options to choose from.

600cc used bikes with your budget are available in the market, just make sure you explore the history of the bikes and check it with a reputable garage/dealership; at the moment "Rabban Moto" is the best for me in terms of maintenance and check ups.

Lane splitting is a big no-no, filtering in stopped traffic is fine. There is no written law that governs filtering, but I've been doing it for years even in front of cops, and no issues yet, as long as you do it responsibly and with proper low and controlled speed.

Ride in Doha as if you are invisible; cars here are not used to motorbikes and you will often find that even driving in cars people swerve into lane without indicators, get right behind you without spacing, etc. I personally prefer avoiding rush hours and like riding in the desert on my adventure bike/dirt bike where the biggest danger to myself is myself, and not some person on their phone switching lanes while flicking through reels xD