According to the description Crush is a psychic bolt. Now following the rules for psychic bolts a target may dodge them as usual. Does this mean that the target can first oppose Crush with its toughness and in case of failure additionally try a dodge? If yes, is this intended?

You're reading it correctly. The power has the Psychic Bolt attack type and thus can be dodged. Other powers don't have this - see Haemorrhage. It is an opposed check with only an effect, no bolt attack.

But isn't it weird that Crush is an Opposed Test AND a bolt which may be dodged? Haemorrhage is just opposed like you said and for example Smite is just a bolt which may be dodged but not opposed ...

Would the Dodge not come first as the Reaction is to the Attack and not the Opposed Test?

I think they're going for a thematic reasoning behind the Dodge/Opposed Test Powers. If you read the description of Smite for example, its kind of obvious why they've allowed it to be Dodged; whereas Haemorrhage makes it clear its a direct physiological effect hence the Opposed Test.

Arguably not the greatest reasoning for deciding what can and cannot be Dodged.

Crush's description says if they fail the test, they suffer the attack. The order is pretty clear, even if it doesn't make much sense on the face of it.

Fair enough, I hadn't read the description properly.

It is not supposed to be a bolt. They copied the power from Only War and there it is clear that if you lose the opposed power test, you just suffer the damage. So I am relatively certain FFG just messed up during copy and past and you are not allowed to dodge it.