You don't always need 'of,' generally, if you don't have 'of,' then the subsequent thing will require an article, unless it's a proper business name. I don't know if this is 100%, but it would be a general rule. There's also some flexibility. On kind of a strategic level, you're probably safer including the 'of' for 'inside of.' In cases where you could get away with not using 'of,' including the 'of,' will sound fine too.
For your example, it's okay, but more naturally, I'd say:
"If we go inside of an operating room...." I suppose in Russian it'd be like: 'Внутри операционной комнаты.'
"If we go inside one of the operating rooms..." Including the 'of' sounds fine here, too. 'Внутри одной из операционных комнат.'
Phrases like "one of the..." functionally are very similar to the article "a." I think it's similar to the 'один из них/одна из комнат' in Russian, which roughly approximates the indefinite article in English. It's indefinite because it's not a specific operating room, but it is from a specific set of operating rooms...i.e. the ones in that particular hospital.
For business names, you can often say it either way, so again, to be safer, you can default to using 'of:'
'Inside of IKEA/inside IKEA.' 'Inside of Walmart/Inside Walmart.' All of these are fine.
Really, in English the 'of' signals a possessive or genetive construction, so it's fairly similar to Russian, I think.