is there a technical possibility to be alerted by email or SMS as soon as a freely definable address receives or has sent a payment,
Yes it's possible. Most web wallets do this by default, but they are not advisable to use for storage.
This would be a bit more difficult to do on a non custodian wallet like electrum.
Do any of you know such a free service on the net? Or maybe the technical implementation of it so one could fulfull this task maybe by running his own full-node and block explorer?
There are some scripts for this around the forum, but I would not recommend any, as I've not used them and you will need to be security conscious.
To get real time alerts for transactions electrum would need to be active 24/7.
You can of course try to create a plug in yourself by connecting to a block explorer and linking an address.