You will need to be sensitively dressed for the Museum of Islamic Art, as they do turn people away, even men. I've been asked to cover up in there myself, and as a Qatar resident I usually dress conservatively anyway (my slightly shorter than normal short sleeves weren't acceptable, apparently).

The heat will be harsh, but at least there is that hop on hop off bus these days to take you around all the main sights. Save outdoor things like Souq Waqif for the evenings.