Now, I am lost at how to move MNE tokens out of this wallet. Swap is available only in small quantities and it is pricey at that. Maybe I could send them - but where then - is it possible? Would a new address be a "normal" address and could be the token be traded from there. On the other hand, if I would like to stake the tokens could I be able to claim this new address as genesis and upgrade it to level 2 or level 3?
No, you can not. The only thing you can do is upgrade your address to Level 2 or Level 3 to begin mining MNE tokens. When (if) you upgrade your address to level 3 you will have the option of mining MNE tokens at a specific rate. You can get about 2 MNE tokens per day per genesis address (it will take about 50 years to receive all 32000 MNE tokens). There is no way for you to trade or transfer 32,000 MNE tokens that were airdropped, but only newly minted (mined) tokens.
You can learn more about upgrading your genesis address
here.