Can I use my UK Mac Mini (Mid 2010)  in the US?
Yes it is safe. You will need a US plug adapter or a replacement cord, but the mac itself is good to go.   

Reading through the specs it states:

Electrical and operating requirements
Line voltage: 100-240V AC

The power supply can handle both the frequency and voltage changes. 

And for the question about buying a power cable or use an adapter -> Up to you I did use both and didn't have any problems. I'd go for a US power cable since that is less clunky, if you stay for longer. If it's only for a few weeks I wouldn't bother buying a cable.