That sounds like a salad made of cheese, not a salad with cheese.
If I knew better a definition/description of what a "cheese salad" is, I would be able to tell whether it could be translated as салат із сиром or not O_o
When you say with blank you add a an "oм" at the end of the word? For example "чай з лимоном" instead of "з лимон"?
In short - yes. З котом, з лимоном, з татом.
The word "with" (з or із) is followed by the Instrumental case. For some masculine nouns the ending is "ом", like in the examples above. Could also be -ем. For feminine nouns one the most common is -ою: з машиною, з мамою.