It all depends on wich games, the United Kingdom uses the PAL system, the only way to know is to try it.
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Yes because the DS is not region locked so unlike the other consoles where you can only play PAL region games (Most of Europe, Africa excluding West Africa, Asia, Australia and New Zealand) in a PAL console or NTSC (The Americas excluding Brazil, Argentina and Uruguay that use PAL and Japan) games in a NTSC console.
No you cannot play games on consoles made for different regions. You can only play games on the consoles made for a specific region. If you attempt to play games cross region it will simply not play.
It will work if you can get a UK power adaptor for it, and if you have a NTSC compatible TV, but you will only will be able to play games from the US, not ones from the UK.
Games for disc-based consoles are region locked. You can't play a Japanese version of MGS4 on a American or European PS3, for example. Cartridge based consoles (especially portables) do not have region-locking. So. Either get an NTSC PlayStation or scam someone else to get your money back. (Can't believe that someone would not actually be aware of region-locking... sigh...)
No the discs are coded by regions NTSC being the US PAL being UK and Europe etc. JAP being Japan and so on..