A writ of habeas corpus (assuming you are filing it in the in the same appellate court as the appeal) serves a different purpose than an appeal. With an appeal, you can only address the issues that appear in the record on appeal. If you are bringing up something that is not within the appellate record, then a habeas corpus is appropriate. Another function of a habeas petition is where you brought an appeal, but failed to raise an issue you wish the appellate court to address that is contained in the appellate record so that you you can then challenge the conviction/ruling in federal court. This is familiarly called an "exhaustion petition". You would not file a habeas petition with the same court raising the same issues you raised in the appeal. If you are going to federal court, then you would do that by filing a writ of habeas corpus.
You do not generally need to request an impartial judge. That is required by their ethics oath. There is a lot that could be explained here, but without knowing your particulars, that is the best I can do. Many lawyers here on Avvo offer free consultations. It might be in your best interest to consult with someone or even pay someone to do this for you.
If your habeas wasn't decided on the merits, you can probably file it now that your direct appeal has been denied.
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Without knowing what ruling(s) were challenged on appeal, it is difficult to say if the habeas petition would be made moot by the ruling on the appeal. If no ruling on the appeal renders your habeas petition moot, yes you certainly can file a new habeas petition.
Based on how you describe the court's ruling on your first habeas as being a collateral attack and that the appeal should have done that, it seems a new habeas on different grounds would be fine.
There is no special request you need to make to have an impartial judge rule on your new habeas.