I don’t remember the exact rules of the story itself and how each Beatrice is a personification of them, but I can try and answer the other questions to the best of my ability for you. (Though as to moving topics, I’m not sure myself.)
The Beatrice revived is, one could say, a Beatrice who is like an innocent child (sort of like the child Kuwadorian Beatrice at the start of EP3). I’m not sure if you’ve read EP7 or 8 or not, so I don’t want to go into too many details, but you could say that he brought her back to life as essentially a new person, somewhat kind of like Victor Frankenstein making a creature, but Battler was doing it for a far less sinister intention, and most likely brought her back with the feelings of love the original Beatrice had. I don’t recall Chick Beatrice ever being referred to as the older one (she calls the longer haired one Older Sister), but Elder Beatrice represents the legend of Beatrice, which has lasted longer than say, 18 years, on Rokkenjima, and Chick Beatrice is younger due to Beatrice’s love developing later than the legend. So Chick Beatrice is not the same person as the one we’ve seen through EP5, she died and gave up on Battler (how she died).
Beato shows her love by tormenting her crush, how tsundere (and it’s why Battler outright says “Just write a letter next time”).
I can explain more if you’ve finished EP7 at the least, but if you haven’t, I don’t wish to get into details.