I am envisioning the 'bog' being the toilet, while the 'loo' is the room with the toilet and usually at least a sink, if not an entire body-cleansing unit, as well as the toilet. I'm not sure that this conceptualization is askew or not.
The loo is the bowl + seat + probably the cistern. So is the bog. So is the toilet (you can sit on the toilet). So is the WC. All of these can apply to the room as well, via what is probably synecdoche or metonymy of some degree. Public or school/office rooms with more than one bowl are generally called "the loos" "the toilets" or "the bogs" but in a house with just one in the room, that wouldn't apply.
"Loo" and "toilet" are commonly used. I've hardly heard "bog" since school, or "WC" outside old buildings or stories.
A bathroom is a room with a bath in it. If it only has space for a shower, you'd probably call it a bathroom. But either way "bath" does not refer the room unlike the case with "loo", "bog" or "toilet". And in Britain, a room with a loo and no bath or shower is NOT a bathroom - I guess in America it is?
Americans seem incapable of using any but the most obtuse usage. We will ask for a 'bathroom' when we have no intention at all of bathing. I guess we just assume that if a room has a bath, it has a toilet. It usually applies in the US. Heaven forbid that one ask for a 'toilet'; that might intimate that one sheds waste. We can't have that.
We also have all sorts of euphemisms for free-standing toilets like those used at construction sites or major athletic gatherings. They are 'johns', or 'cans', or 'potties' (porta-potty is a big fave), but few actually refer to them as 'toilets' despite most directional signs, if beyond basic symbology, points to the 'toilets'. You should see how a segment of the population reacts when there are different symbols on the doors where there is more than one toilet on site....it brings out the cultists.
We used to have, and probably still do have lavatories, conveniences and WCs scattered around rural towns and railway stations. I once saw a public lavatory that was closed for repair with a sign on the door saying, "Sorry for the inconvenience." I thought that was quite appropriate.