#cogwork grinder

No.

First of all, cards you remove from the draft with Cogwork Grinder are removed face down, so none of their characteristics can be seen by any card. Additionally, that card was removed by a card named Cogwork Grinder, irrespective of what names Cogwork Grinder might currently have on the battlefield.

You have to have already have drafted Cogwork Grinder for its ability to be in effect, so it can’t exile itself as you draft it.

Cards you remove with Cogwork Grinder aren’t in your card pool, so you can trade them as long as they’re not going to be part of a redraft at the end for prizes for the draft.

I can’t find a ruling about Cogwork Grinder and DFCs, so I think that’s up to you and your playgroup. My instinct would say it’s fine to remove them from the draft.