Blank Text Message from random number - is it a virus?

Is it possible that this is a virus?

Of course it is possible. Technology is beat. But what actually matters is whether if it is probable or not.

A text message (if it is an SMS) has to travel through the cellular network of your service provider and therefore the file types you can send are object to their limitations. Such messages can include media files(MMS) but not apps or executables. Instead, it might contain a link which'll lead you to download a malware. But an empty SMS on its own to infect a phone, sounds pretty NSA.

How can I check to see if my phone has a virus?

You can use antivirus apps but their reliability is up for discussion.

Is there a way to detect an intrusion attempt so that my phone will tell me if something is happening that shouldn't be?

Again, some antivirus apps like Avast warn you whenever an app that is malicious is being installed. But there are ways to outsmart it too.

Nonetheless, for your specific case, relax. I really don't think you are at risk.