I think there's a bit of linguistic confusion here. Although combination fridge-freezers (i.e. two compartments, one above freezing and one below) are common and we often call the entire appliance a refrigerator, that's more just a simplification. We'd never refer to the freezer compartment (or a standalone freezer) as refrigerating things; it freezes them.
So if someone says "put it in the fridge" or "refrigerate that" or "keep refrigerated", they mean the fridge, not the freezer.
It's rare for things to have more specific refrigeration temperature requirements than that. You can't really adjust your fridge much; it needs to be 40F/4C or lower to be safe. In practice that means you aim for more like 35F/2C so that if some parts of the fridge are a bit warmer it's still safe. That means that even if you wanted to keep something slightly warmer, the warmest you could really keep it is 40F/4C. If you really want to store a food at a higher temperature, you need a different solution; this is why people store potatoes in cellars, for example.
As for what things can be frozen if desired... it's hard to generalize a ton. The main issue with freezing is that it can mess with the texture of food. Fresh fruit and vegetables will generally lose their structure when frozen, so if they're subsequently thawed they'll end up mushy and leaking juice, making them only good for cooking. Things with less structure like meat, dairy/cheese, and a lot of cooked food tend to freeze well.
So your cheese and hot dogs will be fine frozen, but if you have a lot of other things you're interested in freezing and you don't have an intuition for what will and won't work, you're probably best off googling each one.