Russia's history of having a disorganized fighting force (opinion? If so I apologize)

Its definitely an opinion. An opinion that has a lot of evidence to challenge it. I wouldn't say its wrong in an absolute sense, but its not entirely right either. Perhaps its safer instead to say that the Soviet military had visible periods of disorganization, rather than frequent periods.

how impressive was the Soviet Union's army?

Oh Goodness, this varies greatly over an incredibly short span of time. I'm going to need help with this; /u/elm11?

The fact remains that the professionalism, level of training, availability and quality of equipment is simply all over the place between the period of 1920-1945, to give a very rough time frame. The 30s are generally viewed as a theoretical and technical 'Golden Age' for the Soviet Union's military. Officers and Ordinance were very much on the cutting edge, and the Soviet's development of Operational Warfare, industrial mechanization and the modernization of their tank fleet (including, famously noted by Hofmann, the buying of the Christie suspension system) arguably made the Soviet Union one of the most advanced mililtaries in every sense of the word in the early 1930s.

This changed sharply for the worse by the end of the decade for a plethora of reasons that I'd much rather Elm explains if he graces us with his presence.

Depending on whom you read, the Soviet Union's military wouldn't recover from the political and military disasters of the late 30s and 1941 until 1943 or 1944. I am truly speaking generally here, theoretically, industrially, operationally, organizationally, it took a significant bulk of time and 're-learning by fire' to achieve something resembling their pre-war capabilities.

The most significant difference would still be visible in the level of training; constant fighting on a broad front, bloody manpower losses in 1941 and 1942 and necessity for replacements meant that the level of training would not be anything approaching what would be capable in a peace-time regime. If I had to give an educated guess to what is arguably a subjective question, the Soviet Union hit its 'peak' just prior to the start of Operation Bagration in summer of 1944. In terms of organization, available manpower, level of training, quality and availability of equipment, they were undoubtedly at a peak-period in military capability. The exertion of destroying the German's center during Bagration and the last sharp drive from the Vistula to Berlin drained the RKKA more than some might initially suspect. It ended the war a victorious but tired organization; the equipment may have still been there in the same great amounts, but the manpower reserves were beginning to be stretched thin. Losses in the twilight of the war also once again diluted what had been a growing population of veteran outfits within the RKKA. Finally, war-weariness gripped basically every combatant in the immediate post-war era, and that has notable psychological and physical affects on a military.

I'm assuming OP means the conventional military forces when he says no nukes? Honestly it's hard to say since the only real war they fought besides ww2 was the Afghanistan conflict. But, during the Arab/Israeli wars of the '60s and '70s the Arabs were drilled in Soviet tactics and strategies and were given Soviet equipment. Judging by the outcome of those wars, I'd say the Russians probably wouldn't have faired much better if put in the same situation. The Soviet military doctrine was extremely rigid, and did not allow for any flexibility or deviation, and as such it was incapable of taking advantage of opportunities. There was a particular instance during the Yom Kippur war where a Syrian force took a ridge overlooking an Israeli town which was undefended, but because they did not have express permission to advance, they simply held their ground. The town was a staging point for the Israeli counter attack and eventually the Israelis forced the Syrians back from their position.

Of course the Soviet technology was never quite up to par with the West. Their tanks were not as likely to survive in battle, as they had a tendency to cook off once hit by an armor piercing shell. This is because they store most of the rounds in the hull, as opposed to western tanks storing the rounds in the turret in a protected area called a bustle. They also built most of their vehicles to fulfill a specific purpose and thus had less versatility. For example not until the advent of the Su 27 did they have a multi role capable fighter. Almost every previous aircraft was strictly fighter or attack. Compared to something like the f4 phantom which could fight and drop bombs.