How can I tell whether beetroot is still edible?
Generally, raw beetroot shouldn't become soft. The thing is that there are differing definitions of soft. If it is soft like a tomato is, then it has probably spoiled. If, on the other hand, it is still firm but has little "give" to it, then it's probably alright to eat.

It is the cooking that softens it up, though it can be eaten raw. I grate some raw beetroot into salads and stuff occasionaly, just for fun.

I'm not sure what to say about the smell thing. I think that raw beetroot generally smells a little earthy, especially if you haven't peeled it yet. The very distinct beetroot smell is usually released by cooking.

Obviously this is all a bit vague, as we can't send smells over the net yet.