1) What makes a claim extraordinary?
I think one criterium is lack of parsimony.
Another would be the postulation of something "supernatural" (a term which actually takes comfort in being extra-ordinary).
Another would be anything that is defined as being beyond human understanding etc.
Another term that itself signifies an extraordinary claim would be "miracle".
Also, those claims which are esoteric (i.e. postulate that only a special group can understand them).
Etc. etc. These are just the first that I can give from the top of my head.
Then again, most people will agree on something being an extraordinary claim until they make it themselves.
2) Where's the line between requiring sufficient evidence for an extraordinary claim and requiring unreasonable evidence for a non-extraordinary claim?
I don´t think a clear line can be drawn.
Also, I do not really know what "extraordinary evidence" is supposed to be.
3) How does one determine when sufficient evidence has been given to believe a claim?
In certain realms we have established standards. In others (especially those which are, by their own definition, extraordinary, super-, beyond-, whatnot) we don´t.
What, however, we can - at least to a certain extent - can take a look at: How consistent are a person´s criteria when it comes to similarly exceptional claims (e.g. if a person makes a "supernatural" claim and takes comfort in "Well, it can´t be disproven" but tackles competing "supernatural" claims by demanding proof).
Isn't it all just arbitrary based on our own suppositions and prior beliefs / biases?
I don´t think it is "all just arbitary" - but there´s certainly some arbitrariness involved.
I guess the thing that worries people most is when they seem to agree with a person on criteria for determining "sufficient evidence", and then this person suddenly introduces new, different criteria for a particular claim.