If you knew the note names and some theory basics, you could easily name those chords. Technically they are chords, but two note chords are more commonly known as intervals or as we call them in the mandolin world, "double stops."
Without knowing the song context, I'll take a guess at the chords you have been playing:
xx22 = B-F# = B
xx24 = B-G# = E with no root
xx25 = B-A #= B7
xx27 = B-B #= Care to guess?
xx29 = B-C# = C#7 possibly. This would be the II7 chord in the key of B.
x3510 = F-D-D = Could be a D minor, among others.
x358 = #F-D-C = Dm7
x357 = F-D-B = Dm6 ?
x355 = F-D-A = Dm
I strongly urge you to learn some music theory. There are no new chords under the sun and all you have to do to be able to name them is:
1) Know the notes on the fretboard.
2) Know what notes make up what chord.
Here is a tidbit which can prove to be invaluable: you don't have to have the note which names the chord in the chord. In one of the examples above, I showed you an E chord with no E in it. In context, because another instrument will cover it such as guitar or bass; and because of the range on the mandolin - you can only put four notes in any chord until you start splitting strings but that's a whole nother thang.
On a mandolin, double stops which contain the 3rd and the 7th sound very cool. #For example, in the key of G, play:
x32x = G7 with no G
x21x = C7 with no C
x43x = D7 with no D
Try it, you'll like it.
"I thought I knew a lot about music. Then you start digging and the deeper you go, the more there is."~John Mellencamp
"Theory only seems like rocket science when you don't know it. Once you understand it, it's more like plumbing!"~John McGann
"IT'S T-R-E-M-O-L-O, dangit!!"~Me