Or is there something I am missing here? Do you need some information on past registrations/history which you can't access because the luckybit server is down?
Technically, yes. In reality this is true as well - I verify almost everything on the blockchain and the public information is verifiable without the Bet Browser.
However, the Bet Browser is vital to my job since it contains full and organized data about the individual bet, not just the sending transaction, and acts as a sanity check for me against making a mistake while processing an application. (Yes, applications made with a blockchain link are more difficult to process.) I consider it a core step for processing applications to verify every application on both Lucky Bit and the blockchain; this is actually a preliminary step taken during the background check. This does mean, unfortunately, that applicants have to wait to be verified, since I don't have the related data provided by the Bet Browser to check against. This is especially important now that the minimum bet was recently increased - I can't immediately discern if a 0.001BTC transaction is an eligible bet or not.
Everyone's applications will be processed in order and retroactively applied; this actually creates a very unusual case where a slot might technically be occupied by two people at the same time - applications made before slots were available will be eligible to retroactively take those slots, since they will be processed while the slots were open, meaning new applicants will be qualified for slots that hadn't vacated yet. Confused? Imagine how I feel! The upshot is, if there are enough slots after Lucky Bit comes up, all applicants have a chance to get them - even if they applied before the slot was available!
Don't worry about your bet expiring before the engine comes back; it can't. If you placed a bet that wasn't processed because the engine is down, it will still be qualified as long as it was confirmed (on the blockchain) between 7 days before and 2 hours after your application.