Hello,

Your salary would be around 36,500 QR/month.

You can find a furnished apartment (2 bedrooms) for about 12,000QR/month in midtown. If you go to other areas (not too far), you can get an apartment for about 8,500QR/month.
I can help you with that if you want.
If I were you, I would calculate 10,000 QR/month for home.
(You will have to get a loan as you should pay at least 6 months in advance)

For a family of four, I would think you need around 7,000QR to 10,000QR per month for food and other stuff.
I would think 8,000QR/month is enough.

I don't know about the tax laws in the UK, so no answer to that.

Schools are pretty expensive here (around 12,000 to 15,000 QR per student per year).
In your case, this would be around 2,500QR/month.

You will need some part of the salary for social activities, as it gets pretty boring around here, especially for the kids. Say around 1,000 to 1,500 QR/month.

Cricket and martial sports are familiar here in Qatar, but Rugby may not be.

There are clubs and many parks around, but you only get to see them during fall, winter and spring.

The temperatures get really high in summer (44 oC) and so does the humidity (sometimes 100%)

The common places in summer are malls, and there are plenty of those.

I think that could give you an idea, but I would prefer you come and see things for your self first.

Best wishes...