How can I check my mmr without playing ranked?

There is no way to check your mmr. Some sites give you an estimate, but those sites usually only do it for ranked.

Have you played ranked at all ever?

You can't even check your mmr by playing ranked. If you want an idea approximately what rank you might be, who you're playing against in normals can give you a vague idea of that.

Your MMR is at Bronze 2.