Is it permissible for the Aqeeqah to be performed by slaughtering an animal that was slaughtered in the slaughterhouse without mentioning the name of the newborn, and can it be counted as charity if it is not permissible [as Aqeeqah]? Furthermore, is it permissible to purchase two sheep for the Aqeeqah instead of one ram?

For the validity of the 'Aqiqah (definition) (sacrificial offering for a newborn) or Udhiyah (definition) (sacrificial offering during Eid al-Adha), the intention is required, not the mention of the owner's name or its slaughter at home. The ram is sufficient as 'Aqiqah for your elder daughter. Any type of sheep (male or female, lamb or goat) is permissible for the 'Aqiqah, provided it is free from defects and has reached the age suitable for Udhiyah (e.g., a jadha' of sheep, a thani of goats, and so on). The 'Aqiqah you will slaughter later will be for the younger daughter, as the previous ram was sufficient for the elder one.