Why are prayers not answered despite the supplicant's certainty of a response, and is this considered a false hope, especially after performing Istikhara and experiencing depression?

Supplication (Du'a) is an act of worship that Allah loves, and He becomes angry with those who do not ask of Him. There might be a shortcoming on the part of the supplicant regarding the conditions of supplication, for Allah is wise in His withholding and His giving. The supplicant is victorious in every situation; their supplication will either be answered, or its reward will be stored for them on the Day of Resurrection, or an equivalent amount of evil will be averted from them. Therefore, you must humble yourself before Allah and strive in supplicating Him, and be content with His decree, for it is goodness and wisdom.