I'm not sure Soviet artillery could have done better even with better weather; German guns where pre-registered and well dug in, they would have to be identified before any meaningful counter battery work could be done, and the German artillery inflicted it's greatest tactical and strategic menace in the first 8 hours of the battle; also the weather prevented Zhukov's artillery from being visited by Stuka's and JU-88's which would have played havoc with it's effectiveness
Part of the reason that the Panzers where thrown in piecemeal was due to the terrain and lack of usable roads which forced them to arrive in echelons... I'm only aware of the 5th being seriously understrength before the battle started; the Panzers where certainly able to make their presence felt in the numbers they had
I'm dubious of the breakthroughs going anywhere; the terrain behind the salient was terrible and the spearheads would have all sorts of difficulty resupplying if the Germans held the shoulders in strength and interdicted their corridors with artillery; likely they end up as isolated islands that are eventually broken up by reserve divisions
Large numbers of Soviet guns were set aside for counter battery fire, but were only marginally effective due to weather and communication problems; the former, if removed, would allow some fairly effective fire; on its own Soviet artillery was of very high quality, its major failure during Mars was supporting and communication with the infantry. That problem comes into play only after German defenses are penetrated and the artillery needs to reposition. In the real Mars Soviet artillery after the first few hours was hampered due to logistic problems caused by the confined space of the Soviet penetrations, as it was trying to advance at the same time as 2nd Guards Cavalry Corps and reserve rifle divisions. A wider penetration would ease that problem. But again, communication and cooperation problems would remain, but they wouldn't cripple the artillery without other exacerbating factors. Aircraft is likewise a major threat, but the Soviets did have air units and anti-air forces to support ground forces; not enough to halt attacks, but enough to prevent crippling air attacks. The Soviets learned a lot of lessons at Rzhev from the July-September battles where the Germans made very heavy use of aircraft on the defense. Most Soviet artillery positions were very well camouflaged, and assembly positions were concentrated in forests and other areas of good cover. The Luftwaffe would primarily threaten the infantry and mechanized forces once they were in the open. Again though the Soviets took countermeasures when aircraft and AA wasn't available, such as assembling in forested areas. Winter weather, even without major snowstorms, would also hamper the Luftwaffe; low clouds and fog were the predicted weather for the start of Mars, not heavy snow.
Indeed, but that would little change with improved weather; the Salient's roads were fairly poor outside of a few major arteries, which hurt and helped both sides at various points. While a lack of snow would make the roads slightly more passable, they'd still serve as narrow bottlenecks. Anyways, all of the Panzer and infantry divisions were understrength prior to the offensive; some more than other obviously, but most were not even close to full complement. Most had been seriously depleted in the summer and autumn offensives but hadn't received any replacements, as most were sent to other sectors.
Terrain is always a major issues, thought he Soviets were all around superior at all terrain maneuver than the Germans, particularly if the 2nd Guards Cavalry Corps breaks through intact. Plus, greater Soviet penetrations refers not just to dept but to width; effective artillery would see almost double the width of the penetration in the Vazuza offensive, and a quarter more in the Belyi area. This eliminates the primary problem experienced in Mars, logistic bottlenecks and vulnerability to German encirclement. The former is automatically eliminated by wider penetrations, while the later is eliminated by fewer overall losses and the availability of more reserves. This is especially important at Belyi where an entire mechanized corps perhaps two, could have been held in reserve to repel German counterattacks. Such a large reserve would be more than enough to crush the German encirclement of OTL. In the Vazuza battle the encirclement was primarily caused by the Germans rapidly compressing the Soviet penetration and complete Soviet exhaustion. No actual counterattack like at Belyi was launched.