Not sure if I got Remilia's last lines right, so I'd be happy if someone could check that over. Don't have the confidence to really tackle the other lines, sorry.

In the last panel: I think in case of "私にもできるのっ" Flandre doesn´t speak of herself in the first person.

できる takes the action to be done as the subject, i.e. with が (and occasionally as the direct object, i.e. を, maybe?) and the doer as the indirect object, i.e. with に, so the original translation of 私にもできる as "I can do it too" was right. That's also Remilia speaking, not Flandre.