Is there a way to see my MMR? The simpler, the better.
In the patch 7.07 "Dueling Fates", released on November 22, 2017, the rating system got reworked.
This means that you won't see your rating in your profile or at the Find Match tab anymore.
You can still find both your Solo and Party ratings by opening your ingame user profile, and selecting the "Stats" tab. Both ratings are listed on the upper right side.
Yes, you can edit your profile, so that it is always shown, when you look at your profile.
When you click on "Edit Profile", there are two dropdown list, right next to the edit button.
When there are no numbers, that means that you have not played enough games in either ranked solo or group (in my case, I did not play enough ranked solo, for my mmr to show up)