Talking with my brothers girlfriend last night, who is Russian. Her Russian history book basically gives about a paragraph of attention to the D-Day invasion. Something along the lines of, "After the Russians had won the war against Hitler, the Americans quickly invaded from the West to capture what territory of Western Europe they could."
I know that there is obviously propaganda on both sides of the story but I am left with these questions:
You might be interested in this book, which answers most of your questions in pretty good detail. These answers I'm about are basically lifted straight from it, amalgamating different perspectives of course.
Was one of the victories more critical than the other, in terms of defeating the Nazis?
I don't think that there is any doubt that the Red Army and Russian people played a larger role in defeating Germany than any of the Western Powers. As /u/EtEthosLogos points out though, Overlord and subsequent operations were about liberating France and the Lowlands first and foremost, and defeating Germany second. This may seem strange, but even by June, 1944, it was pretty clear to everyone which way the wind was blowing. By the time that Allied armies reached the Rhine, it was certain. The Red Army was destroying whole army groups at this time. The primary factor that the Western invasion likely played is reducing the available strategic reserve to the Wehrmacht. The last of the last that was committed to the Ardennes Offensive could have been used in the East. Would it have slowed the Red Army? Probably. But I don't personally see how it could have ultimately stopped them.
Could the Nazis have retained control over Western Europe against the Russians after Stalingrad?
Stalingrad could only have ended one of two ways; either the Sixth Army could have been left encircled (as it was), or it could have broken out. However, the serve numerical advantage the Red Army had over the overstretched Axis lines in the South meant that a retreat was in order no matter the outcome of that battle. Ultimately, although losing the Sixth Army was a brutal loss for the Wehrmacht, the Soviet war machine was only just ramping up in the summer and fall of 1942. Ultimately, I don't see how the Germans defend against Saturn effectively, nor how they ultimately reverse the material and manpower advantage the Red Army was building up.
Would Russia have been interested in/capable of invading Western Europe if the West hadn't?
I don't think they would have had to. Once Berlin and the rest of Germany fell, it's doubtful the forces in France and the Lowlands would have continued the fight. Nor do I think the Red Army and the Soviet Union was all that interested in making clients out of states so far west. However, what we might have seen is a Germany completely occupied by the Soviet Union, similar to what we saw in Japan for the US.
What other information will help me understand this situation better?
I recommend you read the book I linked above! :)
Stalin spent a couple of years begging/demanding that Roosevelt and Churchill open up a second front to relieve pressure in the East. One of the reasons that Italy was invaded is that they weren't yet able to invade France, but they wanted to help Russia and appease Stalin. Part of the reason that Eisenhower decided to continue with the Normandy landings despite the bad weather is that if they had postponed the invasion, they would have had to wait several more months for the tides to be right, and Stalin was becoming extremely impatient. So while it's true that Russia's contribution was critical to victory over Germany, they by no means did it by themselves. I'll also bet that her Russian history textbook ignored the mountains of Lend-Lease aid that the Western allies supplied to the Russians.
EDIT: http://scitation.aip.org/content/aip/magazine/physicstoday/article/64/9/10.1063/PT.3.1257#thenormandybeaches-1 Source on the tidal issue. As this article details, the landings required just the right timing of tides, sunrise, and moon phase, which were all only available on June 5, 6, and 7. As for the delay issue, it was a long time ago that I read that, couldn't find a source. However, spring tides occur every two weeks, and half of those are with new moons. The tides would be right many times over a given period, but the optimal combination of tides, sunrise, and moon phase was only available relatively rarely.
Just as an aside, Russians and the Soviet Union aren't interchangable, and it weren't only the Russians who defeated the Nazis in the East.