Fonts: languages and pink background

Please accept my apologies for asking this question. It's been many years that I had to do any DTP, and I need some help, please.

I have an Indesign file which was translated in a translation tool from a European language to a language using cyrillic letters. The Indesign file (I'm using Indesign CC, the latest version) contains cyrillic letters now, but the text background is pink. So, the text is not displayed as intended. The source file uses mainly the fonts Minion Pro and The Sans (Plain Bold, Italic).

I copied the cyrillic file into the same folder as the source file, and also copied the fonts into Indesigns "Fonts" folder, and Indesign does find the fonts, it does not show them as missing (no brackets, no error messages).

So, in the next step, I tried to set the Glyphs (Type/Glyphs) to cyrillic, but apparently there are no cyrillic glyphs in those fonts. The fields just come out as blanks, when I select Cyrillic in the Glyphs menu. The text background remains pink.

So apparently, these fonts not support cyrillic characters, and I probably have to to obtain special cyrillic fonts. What I am wondering is, which fonts should I get that are considered similar to Minion Pro and The Sans (if there is such a thing as similarity) in the cyrillic world?

Please accept my sincere apologies for asking this. I really cannot find the information elsewhere, and am a bit insecure as to where the Indesign/Adobe world stands right now with regard to these issues. Big sorry!

Achim