What is the primary cause of death for small animals in urban and suburban areas?
Cats, windows, and cars appear to be the leading causes of avian (bird) deaths:

 Major sources of anthropogenic bird mortality in the USA [from Fig 2b, Loss et al. (2015)]
Cats and vehicles are likely leading causes of death for many small non-avian animals as well.

Loss et al. (2013) estimated that 6.3–22.3 billion mammals are killed by free-ranging domestic cats in the USA each year.

The Wikipedia article for roadkill provides the following discussion of vehicular deaths:



In 1993, 25 schools throughout New England, United States participated in a roadkill study involving 1,923 animal deaths. By category, the fatalities were: 81% mammals, 15% bird, 3% reptiles and amphibians, 1% indiscernible. Extrapolating these data nationwide, Merritt Clifton (editor of Animal People Newspaper) estimated that the following animals are being killed by motor vehicles in the United States annually: 41 million squirrels, 26 million cats, 22 million rats, 19 million opossums, 15 million raccoons, 6 million dogs, and 350,000 deer.


I welcome others to corroborate these claims with a more reputable source in the comments. (I will try to do so myself and will update as I find them.)

DISCALIMER: I was previously only aware of Loss et al.'s work, so the emphasis of my post is on their papers. I did not otherwise search exhaustively for data. Noticeably absent from my post is mention of deaths caused by pest control (trapping and poisoning). I may update this post with more info as I come across it.

 Citations 
Loss, S.R., Will, T. and Marra, P.P., 2013. The impact of free-ranging domestic cats on wildlife of the United States. Nature communications, 4(1), pp.1-8. 
 Loss, S.R., Will, T. and Marra, P.P., 2015. Direct mortality of birds from anthropogenic causes. Annual Review of Ecology, Evolution, and Systematics, 46, pp.99-120.