First of all you can select YouTube from this smart TV, log in and search using the remote, provided you have internet in your home. As for mirroring from your phone, this is not cut and dry at all for me. If you are able to mirror from your phone then the answer is no, you do not need wifi as your phone will provide the linkage to the internet. I know some friends who are able to mirror from iphones but I have not been able to with my ipad, and for that I am told by Geek Squad I need an apple tv device.
Hope this helps.