A lot depends on where you choose to live. Some flats include electricity and water, others do not. Trash removal is your responsibility - in that, you carry it out to the bin outside your accomodation. Waste water going out, free of charge.

No taxes in Qatar. As for surcharges on compounds I can't comment but can't recall anyone every mentioning being charged extra.

If you have a gas stove you will have to get a cylinder (about 200 QAR) and refilling it costs about 15 QAR.

Car insurance depends on the value of your car - you might check the QIC website, it might have info of use to you.

As for mobile phone charges - check on the QTel and Vodaphone websites to get an idea.