If there is an urgency, a urgency pettion can be moved and/or the case can be got preponed for early hearing. These applications are moved before the same the judge before whom the case is pending. The only crietria is that the OP must be properly served and/or put on Notice about such a move.
There is nothing wrong in moving the DJ for appropriate orders, but it is not a good move and most advocates avoid moving such a applications for a variety of reasons.
The High Court is offcourse the ultimate source of admnistrative control over lower judiciary, but to approach the HC you must have adequate reasons to seek an order to jump the que, though it is not unsusal.
Advocates view will depend upon his standing in the legal fraternity, and how he views the urgency factor in the matter.