Can you use a DNA test to find out if you have aristocratic roots and connections with specific personalities of history?

You can. Do an autosomal DNA test and maybe you will find out about the relationship with someone. Hotch, most likely, you will not find relatives, since Russians are reluctant to test. But you can try. Moreover, it is no longer very expensive.

You can. The genome of the direct descendant of Genghis Khan is known. You can go to China, get tested in a laboratory and be tested for descent from the great kagan. I don't know if a similar analysis is being done in Russia, but my friend, who was treated in China, was offered such a service.