Here is my take on these questions...
But how easily is this recognised by others outside the system? How long will that take?
The adoption of badges may take time within different organizations... I do think once there are a greater number of displayers and the badge becomes more common, they will begin to be recognized. How long, ahhh... the future.
Create a Badge glossary ?
The idea of a badge glossary and badge registry has been kicked around in a couple of discussion forums / threads. Also the idea of an issuer registry also has been kicked around... ahhh... the future. I do think this is an interesting idea, but I'm not a big believer in the restraints glossaries, taxonomies, etc. can create. I really want badge ecocsystems for form and be more folksy. Let them change and emerge with the cohorts of learners.
if you were suspicious about Joe's abilities, you'd have to do that (check legitimacy) about his degree piece of paper too, wouldn't you?
I also see the degree piece of paper as no different than the badge... the nice part of the badge is the ability to look at criteria, issuing institution, evidence, etc. is all baked into the badge. Makes for easy checking. Digital badge also has technologies available to check against forgery... paper degrees don't have that build in.
Can we self host the mozilla badge backback?
At this time you cannot self host a backback... well that's not true. it is all open source so you could download the source code and stand up your own backpack. That would be a lot of work. Just wait, a part of the Open Badges roadmap is the federated backpack; http://criticaltechnology.blogspot.ca/2012/10/federated-backpack-demystified.html
Will there be APIs from Open Backpack to e.g. LinkedIn etc.?
Yes, there currently is the displayer API that LinkedIn (and others) could use.
Can a badge belong in multiple groups?
Yes, a badge could belong in multiple groups within the Mozilla Open backpack. I also believe that earners will put some badges into multiple groups so these common badges are displayed in differerent places.
Do you know of known sites that have display cases for accreditation? No. Good idea!