Translation:My grandmother had three surgeries this year.
British English doesn't uses the term 'surgeries' in this context. Operations should be accepted.
We say operations here. To say she had three surgeries would imply a wealthy woman owning three buildings where doctors have their practices.
Not really. "Operațiuni" isn't part of the medical field, so you can't use it here. And I'm not even sure if you can ever swap the two?
No, "operație" is the medical action in surgery, but can be also a monetary transfer, or an mathematical operation. "Operațiune" can be a military plan or a special kind of banking title.