What subjects do you teach? Many private and international schools seek to employ American teachers for to teach English. There are a few agencies that help to place teachers in schools in Saudia Arabia. Apart from applying to schools directly, this might be the most convenient way of finding employment.
Onto your questions,
1) According to the Ministry of Education, all international schools should dedicate one hour per work to teach Islamic Studies as well as a few other Ministry prescribed subjects. A lot of international schools don't place a huge emphasis on these subjects, so the attention given to these studies will vary between schools.
2) I'm not sure about this one. It might be school dependent.
3) Adhering to prayer times seems school dependent. All schools will have prayer rooms, but their use may be restricted to before and after school, as well as during lunch breaks. Schools like the Al Hejaz International School in Jeddah seem to adhere to prayer times. The weekend in Saudia Arabia is from Friday-Saturday, so Jummah won't be a problem.
Are you looking to teach in a more traditional Islamic school?
It might be worth looking into some private schools.
Either way, see this Expat Arrivals page on International schools in Riyadh, and this Expat Arrivals page on International schools in Jeddah. They give you some leads.