Generally, the Muttawa don't enter hotels or expat compounds, and you should be able to relax a little more there.
In Saudi Arabia, everything runs around the five daily prayers.
Computers, CDs, DVDs, i Pods and other portable media devices have all been seized for inspection from time to time by authorities, and you may lose your device if it has anything deemed to be forbidden on it.
The headscarf is however optional for foreign women, but it is worth brining one along simply to avoid harassment by the religious police or other men intent on causing problems.
Saudi law is generally strict where women are concerned and prohibits women from associating with unrelated men.
The exceptions to the closure during prayer time are hospitals, airports, public transport and taxis, which continue to run normally.
Shopping malls also remain open, however the shops inside tend to be closed. Generally speaking, even less observant Muslims make the effort to attend the mosque for this one.
Many shops open again after this, as the intense heat of the day has usually faded by this point.