reply to your questions:

2- if u have a PC u can buy a pre-paid internet card worth 30 QR only(Z-call) that allows u to call UK or the states for hours.u can get Z-call cards from Al-nijada souq (it's a famus PC minimall near The Souq area).

4- there is a Lady Club Gym in Sports City & another one in Aspire they charge 150 for qatari ladies & 300 for non-Q ladies per month,there are sport instructors who can be with u. I have not join them but my friend did,there's another one called Al-Massa near Ramada signals,but they charge 500QR per month or 50QR daily.

wish u nice stay in Qatar :)