Hi!
I would be very happy to have these two questions answered.

1. How can I be sure that the key I typed for my home Wlan access to my router is correct? How can I change or retype the key in case I want to be sure that it is correct? (Now I've typed it in once for my home network and the 770 does not ask for it again)

2. What does a questionmark indicate when it appears next to the network-symbol (globe)?

I've tried to read some instructions from the Nokia pages and this forum, but have not at least yet found the answers to these issues. Thank you for your answers.

While not connected to any network, select Tools - Connection Manager.
Do not click Select Connection.
Now, press the menu button or click the tab at the top; Select Tools - Connectivity settings.
From there, click the Connections button, click the connection you want to edit and click Edit.
When you get to the key, just re-type it.

The questionmark indicates that you do have a connection of some sort but that it isn't functional (for instance, if you don't have a DHCP server running on your wireless access point, your Nokia won't get an IP address even though it is connected to the wireless network.)

For me, the questionmark showed up when I had a misconfiguration or mismatch between the key I entered in the Nokia and the key in my access point.

You may indeed want to re-enter the key (possibly even on both ends) to make sure it was correct.