Does China subsidize its exports? And if so, to what extent? 

Also, how does the extent of China's subsidies compare internationally?

(I understand this is a broad question but I can imagine that some economists somewhere have come up with some convenient measures or indices for this purpose.)
World Trade Organization (WTO) rules prohibit export subsidies and China joined the WTO almost 15 years ago (in December 2001). As a condition of membership, China had to eliminate export subsidies in the late 90s. So, strictly speaking, China can't subsidize its exports. 

However, Dani Rodrik considers that China subsidizes them indirectly through the exchange rate by maintaining an undervalued currency (here and here). Rodrik adds two interesting thoughts. First, the undervalued currency boosts China's growth and this, in turn, is good for the world's recovery and the alleviation of poverty. Second, China could maintain its growth without trade imbalances if it could introduce industrial subsidies to offset a rising yuan. So, it be would better to subsidise tradables directly than to subsidise them indirectly through the exchange rate.