Is the Istikhara prayer that was performed after Fajr prayer and before sunrise invalid and unanswered, given that it was prayed due to not knowing about the forbidden times for prayer, and will God hold one accountable for this ignorance?

You bear no sin for your ignorance of the ruling. Your prayer at that time is considered disliked by some scholars. The Hanbalis go so far as to deem it forbidden and invalid, even if the worshipper was ignorant of the time or the prohibition. Whereas, Shaykh al-Islam Ibn Taymiyyah holds that you are rewarded for it due to your ignorance of the ruling. Praise Allah for what He has decreed, and do not repeat the prayer during the prohibited times.