You can save money here, living the lifestyle which you described. Salaries vary, so depends where you find work -- at a language institute, in a K-12 school, corporate language program, etc. But you should certainly make no less than $2K, and depending upon your qualifications, I would say $3.5 is more the norm (college degree with TESOL certif).

Contracts here tend to be minimum 1 year (more like 2) but you might find some shorter term work at the language institutes and corp. programs if you look...

Saving $1K/month should definitely be do-able.

You will need a car. Public transport here sucks. The upside is that gas is dirt cheap. :-)

Gym memberships vary -- pure gym will run you less than membership at local hotel where you get pool and beach access as well. Single membership can be pricey -- search online here, I think someone recently posted rates for local hotel gym memberships.

Kayaking...have never seen anyone here do that. Most folks here scuba dive, sail, stuff like that. There is a cycling group, loads of runners and folks who do biathlons, etc.

Hope this helps. Doha is a great hub to travel around from!