Would it be possible to make Trophys / Achievements from bitcoin experience One Trophy could be: "First output signed"
And many more milestones on the road to the more advanced usages of the network: "First multisign/multispend", "First completed ANYONECANPAY transaction"

Would that be possible?

So the user would sign a message with the address used to do these actions, and send it to a "bitcoin trophy" website. Then parsing of the blockchain would pick out the events that awards a Trophy for that user.

How can anonymity be conserved, so that a user would feel safe to collect more Trophys?
Yes, that could be done, if you use Tor to connect to the website and if you don't spend the coins from those addresses at places where your identity could be tracked (i.e. tumble them first).