No. You cannot file a writ petition against any Court. & both are not possible.
Because, when the High Court or other any court performs the judicial functions / duties, it won't be treated as `STATE' under Part III & IV of Constitution as interpreted by Supreme Court.
Since, FR are guaranteed against `State' , you can’t file a writ application against High Court Order. You can only appeal to apex court.
Suppose, if it is allowed, the maximum writ petitions are only filed against courts for their delay in justice as the Speedy Trail ( Speedy justice ) is Fundamental Right.
To escape such problems, the Supreme Court might have interpreted that `Courts are not a state when performing judicial functions'. Otherwise, i don't find any logic / justice to say a constitutional body is not a state for the purpose of Part III & IV Constitution.
It is not out of place to mention here one more example about our Supreme Court :
Recently Delhi HC held that O/o CJI comes under RTI Act, 2005 as a public authority.
But what will happen in future, Do you guess ?
One fine day you will find
This matter will go to Supreme Court on appeal and certainly they will deliver a judgment stating that O/o CJI would not comes under RTI Act 2005 . In fact, our CJI has already said this in informal meeting so it is clear that they are just waiting to see the appeal on their table to deliver the judgment on the same lines.
Finally i want to say
Boss (Supreme Court ) is always right because he is the final (supreme) authority .
Note : Anything unwarranted in this, please excuse me as i am an infant (law Student ).