No the wireless will no longer work if it is setup in bridge mode. Once it is in bridge mode it acts only as a modem, everything else is disabled. And furthermore, once it is setup in Bridge mode you cant even login to the modem to check SNR levels.
I don't think you can change the DNS server if you use the bundled router.
I cant stress how much the bundled router has been stripped of features. The only thing you can do is port forward. You cant setup static routes, use dyndns services, setup a VPN server, or check bandwith usage.
What gets me is that nobody has complained about this. I mean you would think the people upgrading to the Lightining service would be considered Power Users but when I talked to the tech he said that he has never been asked to setup the router in bridge mode before. That is mind boggling.