Well, but how does it find out it's [www.example.com's] MAC address needed for 802.11 data link layer?
Your computer doesn't, nor does it need to do so. Since the MAC address is only used within the same L2 network, when you are sending traffic to a different L3 network, all it needs to know is that www.example.com is on a different L3 network and how to get to that L3 network.
Generally this is the default gateway for most devices, and as you point out, your computer already has the information for your router in your ARP table. The router will adjust the L2 information before it passes the traffic on to the next L3 device.
Why is it even necessary to add the MAC address? The IP address will already uniquely identify it, won't it?
No, IP address are not necessarily unique. They must be unique on the local network, but they can be re-used. This is one of the reasons that NAT exists.
Further, Ethernet is only one L2 protocol. IP is designed to work without concern for the underlying protocol, so you could replace Ethernet with something else (like Token Ring or Frame Relay) or you could create your own L2 protocol if you wanted. IP wouldn't care or be bothered by the change.
And how do routers look up the IP adress, if it is hidden beneath the data link layer? I don't really get it...
The L3 header is not "hidden" by the L2 header. The L2 header doesn't change the content that becomes the L2 payload, it is simply pre-pended or added before the L3 header. Many devices will strip off the L2 header before L3 processing, but even if this isn't always the case it doesn't obscure the meaning of the rest of the message.
Think of it this way, let's say I send you a message that said, "How are you today?" This passes through a translator who changes it to, "Hello, how are you today?" Can you still understand my original message even though a new "header" was added to it? And that additional header may be removed before final delivery to you.
But what MAC address will be written inside that field between my router and the destination router?
When your router sends the traffic to the next router (assuming they are using Ethernet), it will use the MAC address of it's interface as the source MAC address and the MAC address of the next router's interface as the destination address.
There are probably dozens of questions on this site that focus on different aspects of these questions. Please feel free to use the search function to find them. Here are a few to get you started:
And since you touched on 802.11: