Is it sufficient for purity and prayer to perform istinja' after a wet dream and the involuntary emission of semen, or is ghusl obligatory, requiring the repetition of prayers, taking into consideration the cold weather?

A wet dream is not considered a sin and does not necessitate seeking forgiveness. If the emitted fluid is semen, expelled forcefully and with pleasure, and possesses the characteristics of semen, then ritual bathing (ghusl (definition)) is obligatory, and the prayers offered before performing ghusl must be repeated. However, if it is pre-seminal fluid (madhiy), then ghusl is not required; rather, purification with water (istinja (definition)') is necessary, and the affected area must be washed, and the prayer remains valid. If you are uncertain whether it is semen or pre-seminal fluid, you may choose to apply the ruling of either one.