Is wasabi considered to be spicy or to be a spice?
In my book, this is pretty trivial. Wasabi is absolutely a spice - it's something with a very specific flavor, derived from a plant, that can be used in fairly small quantities to add flavor to something. 

It's not spicy (spicy hot, piquant) in the normal sense, though. It doesn't contain capsaicin. It is hot in some sense: it contains allyl isothiocyanate, which we obviously have a very strong reaction to. This is the same compound that's in horseradish and hot mustard. The reaction is very different from capsaicin, though. Capsaicin causes you to register heat at a much lower temperature than you normally would, so you're actually feeling heat, like you would if you were burning your tongue. It's quite literally hot, as far as your body is concerned. It's also an oil, so it can't be washed away easily with liquid, and the burning tends to linger. Wasabi, on the other hand, doesn't produce an actual sensation of heat, you feel it mostly in your nasal passages, and can easily be washed away with liquid, so it tends to be a brief sensation. So sure, "hot" is a reasonable way to describe it, mostly because we don't have a word for the actual sensation, but it's definitely not the same thing as a hot pepper.

I'm sure you can find plenty of people who would say that these are just two different kinds of spicy hot (piquancy), but arguing over definitions isn't going to get us anywhere. The important thing is that there's a fundamental difference, and there's no way you could substitute one for the other. If you want to understand why I think this (and it's not just a personal definition), go to any recipe site, search for "spicy", and see how many things with wasabi/horseradish/mustard supplying "spiciness" you find.

Edit: To stave off further debate in the comments, let me just repeat: arguing over definitions is not useful. There are probably a lot of people who think "spicy hot" should include this, and a lot who don't. I wouldn't generally expect anyone to think of wasabi without context when you say spicy hot, but you're welcome to use the terms however you and the people you talk with understand them.