Can I make a back-to-basics networking suggestion:
Assuming it's an IP camera you're configuring for the GMX email, and assuming there may be some unknown network access rule getting in the way :
Temporarily unplug the camera from the POE switch, plug the PC in its place.
Set the PC IP address to match what the camera has, and also set the DNS targets to be the same as the camera.
At a Windows command prompt, verify name resolution as follows :
ping mail.gmx.com
and confirm that it resolves to an IP address and also gives a ping reply.
So far, so good if it does.
If name resolution doesn't work, try
nslookup mail.gmx.com
If that fails, try
nslookup mail.gmx.com 8.8.8.8
If that does work - there is a name resolution problem with your DNS target, presumably your router.
Re-try the email test in the camera with Google's 8.8.8.8 address for the DNS server.
If name resolution does work, try a ping to one of the GMX mail servers, for example :
ping 212.227.17.174
ping 212.227.17.184
If that does not work, check routing with this :
tracert 212.227.17.174
and wait for at least 10 hops to be tested.
If that fails, there may be an outbound block implemented in your firewall rules.
This should give some useful info about the network environment that the camera sees, although there could be some differences due to the PC being the test host.