I don't know if there's any way to know if the cards are new for the exploration part, but they will only be cards that are available via other exploration methods. You will never be able to 'miss' a card in the sense that it will always be available elsewhere, either on the stage it's from if you've not gotten that far or via Girl Hunt Quest.
As for slayer cards, every event gets new slayer cards. If the event has two phases, a new set of slayer cards will be introduced in the second phase, and these cards will be better than the first phase cards. Also, the newest 'alpha' cards will become the 'beta' cards in the next event or phase. This means if you get an alpha slayer from the event gacha during the second phase of an event, you will be able to use that card for the event and as a beta slayer for the next event.
Lastly, all events have an N-Slayer that can be obtained by the normal N-gacha. As with other slayers, if the event has two phases, there will be two N-Slayers, one for each phase.
Typically, to get a slayer card from the Event Gacha, you need to be VERY lucky or have at least several event gacha tickets. Typically, you need at least five to get lucky. The N slayers can be an absolute BITCH to get. Sometimes you get them on the first couple rolls. Sometimes you have to spend HUNDREDS of N-gacha tickets to get her. For what it's worth, I find that new N Slayers have a higher tendency for the upper left box, but maybe that's just for me. They can appear in any box if they show up at all.