Is it permissible to see only a picture of the fiancée's head? Is Istikhara prayer obligatory even though she is righteous and desirable? Is it obligatory to thank God for her engagement before the marriage is finalized?

You are permitted to look at the fiancée herself. It is sufficient to look only at her face and hands. Some scholars have allowed looking at what commonly appears during service, such as the neck, hands, and feet. The suitor may repeat the gaze until the purpose is achieved. If the purpose is achieved, he must refrain from looking further, due to the encouragement found in the Sunnah regarding this matter. It is also commendable to perform Istikhara (prayer for guidance) in every permissible matter. A righteous wife is among Allah's blessings upon His servant, and she is a source of happiness for the son of Adam, for which one should thank Allah for this blessing.