Yep, you can. Still in the 20s, reaching 30 at mid day at the moment. Locals seem to have got the winter jumpers out, but not seen many Europeans in denim yet.
I only visit Bahrain, but City Centre would have a good selection of places.
Yes it's allowed here. Jeans are widely available as are everything the shopping mall is just like one anywhere with the usual stores Zara H&M, top shop GAP etc