Here's my characters. OK, I did more than just one character.
As you can see I have one Adept transferred to SP and another at lvl18 then sentinel at lvl20
rest of the characters were unlocked somehow cause I don't remember playing them and definitely not Soldier cause I don't like this class since first game. 
Is there any way to check how many hours you've spent in MP ?

In fact, yes, although the results were, I confess, a little confusing.
I will spoiler your N7HQ, assuming that I have found the correct one.
Edit: Removed as it was the incorrect profile.
I can't be sure if that's actually you. If it is, and you are uncomfortable having a public forum link to your profile, let me know and I will edit it out of my post.
The odd thing about the link is that it seems to indicate that you have 0 games played, and 0 hours of play. It might have something to do with whether you checked the box that enables the "feedback" to EA's servers. If it is unchecked, your data would not be uploaded.
However, I can explain the rest based on the screenshot you provided. The characters you didn't play that are above level 1 would have received their experience from "character cards" that you received in packs, which is automatically added to all characters in that class.
You have one promotion on your adepts (that would be the one you used for War Assets in the single player campaign) and you most likely played the Sentinel (it's fairly difficult to reach level 20 using only character cards if you've only played about 10 games, give our take.)
The unlocked characters are colloquially referred to as the "Vanilla Kits" by the MP community, and they are automatically unlocked when you start playing.