Download and use 5.6 utility for Lion.. v6 in Lion can run alongside 5.6 to provide pretty network pictures.
http://support.apple.com/kb/DL1482
You will want to download your local version in Dutch??
So the cabled computer has no problems.. definitely a wireless issue then.
Could a '.' in the name be a problem?
Yes, if you are going to fix names fix them properly. No spaces no non-alphanumeric.. it is not that some special characters cannot be used.. it is to reduce the chance of problems. Wireless names (SSID) does not contribute to security at all.. except that it is good to use a name different to standard. ie if all brandx modems and routers use wireless name brandx it allows people to precompile lists of brute force attack. If the name is different they need to create it on the spot.. but in neither case is it a big issue.
I name the TC.. TC3 for example.. call the 2.4ghz wireless TC24ghz and the 5ghz TC5ghz. Just use something that makes good sense to you so you can identify it as your signal not your neighbours.
BTW, does that mean I'll have two networks?
No both 2.4ghz and 5ghz will share the same ip range and can talk to each other.
It is just like having two different ethernet cables. The problem with wireless, is it is single channel half-duplex.. and like the old thin ethernet, everybody shares the same band.. and divides the bandwith. Using two different bands allows two computers to talk at the same time.