It's good to see more and more new faces around!
1. Fairy Godmother cards are for boosting the stats of an attack / assist card of your interest. To use it, go to the カード合成 menu, set the card you want boosted on the left box, drag the FGM card to the right box, and hit the 合成する button in the middle. The stats of the card on the left will then be boosted depending on the rarity of the FGM used: 1.5x for R+, 2x for SR, and 3x for SR+ (the boost factors are written on the face of the FGM card).
2. The cards with red borders are Ultimate Cards. They can only be made from SR+ cards that have reached High Purity (rainbow background). The advantage of Ultimate Cards is that they can be powered up to permanently increase their stats, up to 60000 / 60000 for attack cards and 60000 / 570% (if I remember correctly, though I'm likely to be mistaken) for assist cards.
3. Card abilities are just animations of the cards, really. They serve almost nothing other than to make the cards look awesome in action.
I guess that's it. If you have anything else to ask, feel free to do so! You can also visit the
walkthrough thread for general discussions on the game itself.