The apartments outside of a compound (that I know of) DO NOT have any other facilities other than the apartment itself and MAYBE parking. They do not have pools nor gym facilities. Certainly there are public pools and gyms that you can access for a nominal fee.
Most compounds will have a pool and a gym. Some have additional facilities like a small grocery store, laundry/dry cleaning facilities, restaurant, bar (some even serve alcohol - there I said it!! Haha)
Apart from a general statement that you find a lot of expats living in a compound, there is not really a specific part of town where they live. Keeping in mind that quite a few of the compounds are kinda in the outskirts of the city center.
For internet - if you live outside of a compound you can get internet access by subscribing to one of many telecom companies in town. In my compound I pay SAR500 every 3 months for internet access. Speed is not too too bad - I do not do much downloading of movies although I download songs.
As far as which area of town to live - well I can tell you my personal preference (which may or may not be in line with yours). Even back home I like living in or near the city center. I like the idea of having easy access to things without having to drive and sit in traffic. That is the reason why I chose to live in the center of town. My compound is literally between the Kingdom Tower (to the north) and the Faisaliah Tower (to the south). There are restaurants and coffee shops, shopping within a 2 block radius of my place. Yes there is a lot of traffic - but then again there is traffic ALL over the place in Riyadh!!
I am sure there are sports bars (or close to it) in town - I just have not found them. You will find an occasional coffee shop or restaurant showing football but I have not found a place where there are multiple screens showing different games. I am sure some of the more seasoned expats know where and if such a place exists.
There is one compound that I know of where a lot of Brits live - Palm Court. I do, believe, however that it is a bit difficult to get in and the folks I know who live there went through their employer to get them their unit. Otherwise, you can always be "guested" in to the compound once you get to meet the other Brits.
In terms of security, Riyadh (and Saudi Arabia) in general has a very low crime rate. I am not saying that it does not happen but it certainly is not to the extent of the crime that you read about back home. Having said that I am sure there are parts of town that are more desirable than others. I am sure some of the expats who have been here longer can tell you the areas to avoid. I can say though that there are a few car breaks ins that have been reported. A couple of my friends just had their cars broken into 3 nights ago (they live outside of a compound). They live in the south part of town.
I hope the info helps.