Is this possible?
Do I have to use a wifi shield or can I also use an ethernet shield with a router?
Both is possible, which one to choose depends on experience, budget and space requirements.
Which will be easier?
Typically the version with the external router is easier to debug if something goes wrong, because you can attach a laptop and try to connect from there. Debugging a non-working WiFi shield can be tricky for the beginner.
Is this feasible too?
From a technical view: yes. You have to decide about the security view. If someone is able to decipher your network traffic he will be able to unlock your door...