It is impossible for a “damages award” to be issued by a judge for an initial illegal work action, but I am certain you already knew that.

Decent enough summary. Petitions for rehearing are usually only granted where there is a glaring, straightforward, completely-obvious error (usually of law) by the court (i.e. the Supreme Court voided a law, but the decision relies on the voided law), and even then often not. Rehearing en banc petitions (for hearing by all justices rather than just a panel of 3) are expressly-disfavored and only granted where the justices feel that they should all hear the case in order to maintain the consistency of the court's rulings and/or the case is exceptionally-important.

Both are opportunities to get paid for writing a really-nice brief that won't result in a win. The legal equivalent of a Hail Mary pass. Or this buzzer beater: https://www.youtube.com/watch?v=S818bdKXfrw