1. Is my prayer valid if I pray a missed prayer during a time when prayer is forbidden, with the ablution I performed before that time, despite continuously passing wind? 2. Is the ablution of someone who continuously passes wind invalidated if he performs ablution for the Duha prayer and then the time when prayer is forbidden begins before Dhuhr?

If flatulence is continuous and unceasing, it falls under the ruling of سلس (incontinence). Its owner must perform ablution (wudu') after the entry of the time for each prayer, and he may pray with this ablution as many obligatory and supererogatory prayers as he wishes. The times when prayer is prohibited (أوقات النهي) have no bearing on this ruling. So, if someone performs ablution an hour before Dhuhr, he may pray with this ablution as much as he wishes until the time of Dhuhr enters.

As for praying missed prayers with this ablution, it is valid. You are not required to perform a new ablution until the time of Dhuhr enters, unless your ablution is invalidated [by something else].

Al-Bahuti said: "A woman with prolonged vaginal bleeding (mustahaḍah) and the like, such as one with urinary incontinence, prostatic fluid discharge (madhi) incontinence, continuous flatulence, or a wound whose blood does not stop, or constant nosebleed... performs ablution upon the entry of the time for each prayer, if anything exits, and prays as long as the time lasts, both obligatory and supererogatory prayers. If nothing exits, ablution is not obligatory."

If you perform ablution for the Dhuha prayer after sunrise, you may pray with this ablution until the time of Dhuhr enters, and the time of prohibition has no effect.

And the times of prohibition do not prevent making up missed prayers.

As for the issue of the invalidation of the ablution of one with incontinence: whether it is invalidated by the exit of the time or by the entry of the time of the next prayer, there are two opinions on this matter. It is more prudent to perform a new ablution for the Dhuha prayer and the two Eid prayers.