When the core client runs through Tor and looks for seed nodes. Is DNS still used? And if so, does it go through the Tor proxy too?
Yes, if you configure Bitcoin to use Tor then it will use Tor for the DNS seeds.
A good thing that BitcoinJ does is that it makes sure that it connects to the DNS seeds via at least two(?) different Tor exit nodes. I don't think that Bitcoin Core does this, though it might be a good idea.