While this isn't a direct answer to any of the three specific questions posed, the Battle of St. Gotthard is a pretty good example of how a sudden change in weather could be a complete disaster. The Holy League managed to beat the Turkish army almost entirely because of the weather. The League soldiers had a small advantage in musketry (the Ottomans were notoriously slow to adapt on that front), but the Ottomans had a huge advantage in basically every other category. Even if the number of guns they supposedly brought to the field is significantly greater than what they actually had, they almost certainly had a significant advantage in artillery. They had a massive advantage in manpower. The Turks had the advantage of only needing one command language (and there were already systems in place for integrating Tartar auxiliaries and client state levies) whereas the League had to juggle numerous languages: French, Hungarian, Czech, Croat, Italian, and at least two (probably more) German languages. The Ottomans also had a vastly superior logistical network, though that wouldn't play a huge role in the outcome of any specific battle beyond allowing the Ottomans to concentrate an unusually large number of men in a given area.

All of which is to say that, even accounting for the fact that Montecuccoli was a likely a better commander, the balance of probability says that Ahmed Koprulu should have won the battle. But the weather backfired. When Koprulu's troops initially crossed the river, it was at an unusually low level: his army could pretty much just march across. The Carpathian basin is criss-crossed with numerous rivers where that usually isn't the case; standard Ottoman procedure was to use a combination of purpose-built bridges and pontoon bridges. But since that wasn't necessary, those additions were never built.

And then the river flooded overnight. Suddenly the Ottoman army was cut in half. The League capitalized on this and attacked. The vanguard (largely professional troops lacking artillery support) were effectively cut off. They were cut to pieces. Large enough numbers of them drowned trying to ford the flooded river that Western records of the battle often spend a good deal of time talking about the weather (Sir Paul Rycaut, who was attached to the British embassy in Constantinople at the time, used terms like "deluge" to describe the flood). Koprulu still had a rather huge army even after the defeat, but having lost so many of his professional soldiers, and facing some unusually hostile weather, he ended the campaign nearly a month before the usual Ottoman end date. And the Ottomans were well aware of these limitations: the year before Koprulu had received a letter from Mehmet IV ordering him to end the campaign by mid October at the latest. Koprulu would ultimately capture Uyvar before Mehmet IV's deadline, but the incident shows that the Ottomans were well aware of the problems of campaigning in Hungary when October rolls around. Koprulu's campaign ground to a halt not long after Uyvar fell: after it became clear that it was taking his artillery an entire day to move as far as his infantry could move in an hour, the campaign was basically done (the usual Ottoman expectations suggest a ratio closer of 2:3 for infantry and artillery).

Mehmet IV's caution probably has something to do with Suleiman's retreat from Vienna in 1529. Suleiman maintained the siege into October and when he abandoned it, he forced his army to endure an awful march home. He lost most of his baggage train while crossing the Leitha at Bruck (which isn't terribly far from Vienna; it's still in modern day Austria). In addition to an appreciable chunk of his war materiel, he also lost most of the tents. That may not sound like a big deal, but trying to cross Hungary by foot in the middle of a rainy, cold season armed with nothing but a sleeping bag? That sounds like a special kind of hell.

I am reminded of the battle between oda Nobunaga and Imagawa Yoshimoto. Nobunaga had ~3,000 men with which to confront Yoshimoto. In general, no one even expected the Oda to make any show of force against Yoshimoto's march to Kyoto. So, this leant more than a small element of surprise to Oda's attack.

Once in the field, Nobunaga learned that Yoshimoto's force was celebrating its recent victory (assuming relative protection from their numbers and terrain).

However, there was a violent storm that was accompanied by a thick fog that Nobunaga leveraged to deploy his much smaller force into the Yoshimoto camp. Once the storm broke, Nobunaga launched an assault that quickly sewed chaos in the Imagawa army and the Oda forces were able to kill Yoshimoto. The invading army, a composite of factions, having lost a very unexpected battle and the leader of a tenuous union (the Matsudaira were effectively a client state), the Imagawa were basically defeated as contenders for national power.

Ultimately, this speaks to the lack of desirability of fighting in the rain. Unexpected weather events create unpredictable circumstances and change the balance of power. Whether the weather creates mud, immobilizing a large army, causes a flood, separating an army, or created fog, making it difficult to assess enemy movements, bad weather was generally problematic for organizing larger forces. Generally, bad whether causes a rapid scaling problem that causes real logistical problems for a large force that may not be as problematic for a smaller force.

Edited because I can't type on mobile.