How would I know whether an iMessage has come in for me? I don't have an iPhone. I don't usually have iMessages turned on. But if a friend sends me an iMessage, I won't see it for months, if I turn on iMessages on my MacBookPro.

iMessages are associated with your Apple ID, and as you know, only work between Apple devices. In theory, you don't need to keep your Mac running. Of course, the only way to access iMessages would be when your Mac is powered-on, and then, run the Messages app. In theory, any iMessages you would have received when your Mac was either powered-off or powered-on, but with the Messages app not running, should appear after starting up the app.

You don't need an iPhone to receive iMessages on your Mac ... only if you want to send or receive SMS text messages, on this same Mac, would you require one. Typically, you need to run the Messages app, or have it running in the background, on your Mac to receive iMessages.

Ref: Use Messages with your Mac - Apple Support