Is it permissible to recite Surat Al-Fatihah in prayer with the intention of seeking healing, along with the intention of worship and its being a fundamental pillar?

A specific intention is not required for reciting Al-Fatiha in prayer. The intention for the prayer is sufficient for all its parts. There is nothing to prevent the worshiper from having another intention, such as the intention of seeking healing, when reciting it. This does not affect the validity of his prayer.