No you can't use an Uzbek driving license to drive in Slovakia, you need to get an international driving permit. It doesn't matter if Uzbekistan considers it to be international, an IDP is a different document and is obtained separately. Of course, they still need their original driving license. The US embassy and state department also advise people to get an IDP.
Requirements for tourists
A driving license issued in the United States or other foreign
countries is not valid in Slovakia until the driver does not have an
International Driving Licence.
Exact information is hard to find, but you could be fined 30EUR or anywhere between 67 and 332EUR for driving without the right documents. Note that police in Slovakia are allowed to fine you on the spot (but payment can be deferred).
As for using photocopies, from searching online there's no mention that they're accepted. Making copies is usually only good for backup or to submit forms, I wouldn't risk relying on that in case you get stopped or to try to rent a car.