From what I understand, you do have WLM on your phone. It's built into your messaging hub. It's just not labeled as WLM.
If you click the Messaging tile, swipe right to "online."
Click the ... on the bottom right and choose the icon that looks like a person. This is that "status" icon. If you choose anything other than "offline", you'll be logged into WLM with the main LIVE account you used to set up your phone.
But this should NOT send invitations out to anyone. So I'm not sure what's causing that with your situation.
I've been chatting with Windows Phone Support on Twitter - turns out, one of my contacts received an IM from my MSN Messenger account (strangely labeled as though from my Gmail account, despite my WLID being my MSN email address, and not my Gmail address) with a pr0n link.
So the culprit is likely that my account was phished. Now that I've got a Windows Phone, all my accounts (gmail, FB, MSN) seem to be linked, both on my phone as well as on my Live account.
While I was searching for a way to transfer my contacts from my iPhone to my Windows Phone, I may have clicked on a phishing site - this is the only explanation I can think of.