1) You're probably right
2) Me and others I know have used this and recognize all our likes. There are cases where you accidentally or unknowingly like something which may explain the Likes you claim to never have liked. I've had my FB account since 2006. I don't know about you or others but there's bound to be some likes you don't remember if you've been a member long enough.
3) I agree - I was a little misled at first thinking there would be buttons to unlike pages. That said, and correct me if I'm wrong, there actually is no way to unlike a Facebook page apart from visiting the original page or clicking the embedded Like button you originally clicked to Like it a second time. I don't think this is the developer's fault. This is a Facebook issue and the developer seems to be working around the limitations of "Unlikes".
4) This is incorrect. If you hover over the grayed out checkmark you can click again to unlike. By default all of them should be grayed out and checked anyway since you've already liked them. Hovering over the check mark will show an X. You click the X and it's unliked. It is hard to understand and not intuitive on a number of levels but again, this is something out of the developer's control.
I understand your concern that we should be wary of random sites that ask us to log in with Facebook and you're justified in having reservations but in this case a lot of your concerns are easily explained by the Facebook API itself. This looks far more legit than not. The only sketchy part about it was the "Log in with Facebook" request - but once you get past that you see it's legit.