No.
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Yes, DS games are region free.
If the UK is selected, then it's a PAL region DSi. DSi systems are region locked for certain content, you can only change the country and not the region. The US isn't an available country on a PAL region DSi, it's available on the American DSi.
DSi Ware games and other downloadable content The Nintendo DSi Shop is specific to the region for which the system was manufactured. For example, a Japanese system can download DSi Ware games and other software only from the Nintendo DSi Shop for the Japanese region, regardless of where the system is being played. Nintendo DSi exclusive game cards Nintendo DSi game cards that are only compatible with Nintendo DSi and Nintendo DSi XL (and not Nintendo DS or Nintendo DS Lite) are region locked. This means game cards manufactured for the U.S. market would not work in a system made for another country. Nintendo DS / Nintendo DS Lite game cards Older Nintendo DS and DS Lite software is region-free, so you can play most of that software on a Nintendo DSi and Nintendo DSi XL from any region. It's possible, however, that future Nintendo DS software will incorporate region encoding. In these instances, these games would not work in a system made for another country. Source: http://www.nintendo.com/consumer/systems/dsi/en_na/faq.jsp#vers
Go to Nintendo DSi Shop. There will be games where you have to trade in DSi Points for the game and there are some that are free. Just click on the free game and click download.
Yes, because the Nintendo DSi is not region locked