"Is it correct, that you don't get any wyrdstone shard rewards in neither the "ambush!" nor the "the pool" scernario? "Scenario 1: Treasure Hunt" explicitly says that you get one shard for every counter you posess at the end of the game, but the previously mentioned scenarios lack this note."
We have always played that every wyrdstone counter your band has at the end of the game = 1 wyrdstone shard you can sell afterwards (in addition to what you might find in the normal search process). I hadn't noticed that there was a statement in Treasure Hunt not found in the other two. I suspect it is just an editorial oversight. Frankly, if wyrdstone counters don't equate to wyrdstone shards, there's not a lot of point to those scenarios!
"And does a hero carrying wyrdstone counters earn +1 experience point for each counter (meaning 3 counters = 3 exp?) or just +1 exp if he carries _any_ amout of counters?"
In all three scenarios it specifies "+1 PER wyrdstone counter" [emphasis added].
"Last question, does "simply moving into contact" with a wyrdstone counter mean that you have to end your movement at the counter? Can you e.g. run/charge over a counter and pick it up?"
In "The Pool" you search at the end of your turn, so no running/charging past a shard there if you intend to pick it up. In "Ambush" you start with them. In "Treasure Hunt" it says "Warriors can pick up the counters simply by moving into contact with them." Since it does not specify that movement ends, or that you pick it up in a different phase of the turn, presumably you can move, pick up a shard, and continue moving if you have not already moved your max. Certainly that is how we have always played it.
Slightly off topic, "Ambush is a scenario that in our experience never works well, and we just reroll it when it shows up. If there are three bands, the band with the shards will pretty much inevitably get all of its heroes off the table - there are just too many ways to run. But with 4 or more warbands, the band with the shards is almost always annihilated.