So I looked at the youtube video and the chords on your chord page and they seem correct. I think what you are hearing is the fact that the guitar in the recording is probably playing capo’d up to the 7th fret, or even the 10th, and playing in G position E position which means the chords are correct you just need to know the transposed chords. (in G position the D becomes a G, G becomes C, A to D, and Bm (or Hm) to Em etc…) That said I would play this with open chords on the Mando and it should sound fine.
As for singing with Mando, it will not give you the full back up like gtr but it can work. Advice is look at the pros. There are two basic ways I see the rhythm technique like Tim O’Brien and Chris Thile. Which emphasis the chop or rhythm aspect of the instrument. (see the 2 links) I have put the Thile one up on the forum before, and he is pretty intimidating but fun to see what is possible or unpossible( he sings at 3:58. If you like this stuff just search Thile and Floydfest and you will get a couple more like this.)
Tim O. http://www.youtube.com/watch?v=fMl3xViMvY4
Chris T. http://www.youtube.com/watch?v=ZTis0S9zxXg
The other technique is droning the g string to give you a bit of that Gtr sounding back up when singing. When it comes to that Steve Earl has a great example from about 22 years and pounds ago. This is a great version.