With this setup, neither side is really going to be able to beat the other, so that changes the motivations and objectives for both sides. I'd suspect the war would be relatively short and limited in nature.
Most fantasy settings have pre-modern populations. Countries might have millions of people at most. Meanwhile the US has 320 million people. Drafting 10% of the US' male population would result in an army that could outnumber the total population of the Horde and the Alliance combined, and that's not even touching industry or agriculture. Yes, Azeroth has magic, but this feels like a
WW2 style Japan vs USA situation, where the underdogs would need to have a consistently amazing killrate just to have a chance at winning, which will statistically not happen. Azeroth has zero hope of being actually able to conquer and hold the USA, and carving out even small chunks of the USA would be impractical.
What Azeroth can do is cripple the USA's ability to threaten or pressure them. Their superpowered warriors and wizards could go around
sinking naval fleets, raiding nuclear silos, taking out key enemy personnel, etc. As the USA's military assets are being ripped apart, the USA's real enemies like Russia and China are going to start moving in on the wounded animal, at which point the USA's leadership will have to wake up and realize that they will lose more than they will win if they continue trying to fight Azeroth. So a truce is signed as the USA prepares to defend itself from the other superpowers, and Azeroth moves to a policy of deterrence, having shown to the world how they can screw you up if you mess with them, so no one tries to make a move on them in the first place.
As for the USA... the magic, herbs, wildlife, and technology (mainly Draenei, Titan, and Legion tech) of Azeroth will interest the USA military enough to try to acquire it for themselves. Numbers wise, they have enough soldiers that there is a decent chance that they will eventually be able to capture a sorcerer or an anime swordsman, and then interrogate or vivisect him trying to learn the magic. The megacorporate state will try to exploit the newly found world of Azeroth, hence the initial invasion. Probably turns into Vietnam but 100x worse. As the USA's military gets screwed up and the military realizes they have bigger fish to fry, the war will come to an end. Trade agreements might happen later.
If the forces of Earth invade through a Dark Portal, then no they cannot win. If Earth develops spaceships and actually travels to Azeroth, then it's game over for Azeroth because the Earth forces can just bombard Azeroth from orbit and there is no way the Azerothians can intercept thousands of incoming objects at such high velocities. Maybe the Draenei can create lasers that shoot down incoming missiles and cover the whole world, but a simple solution like bundles of telephone poles will simply have too much mass to be stopped (whether or not the Draenei lasers can vaporize them in time is another debate). Only the Scourge might have a chance against the forces of Earth, if only because they are not dependent on being alive (if all the farms are burned and everyone else dies of radiation poisoning, the Scourge will still be around), and because of their ability to quickly replace their numbers.
The Zerg effortlessly wipe out and take over Azeroth. The Zerg took over the Confederacy capital world of Tarsonis in a matter of hours or days. The only reason they didn't take over the entire Starcraft universe was because the Zerg were controlled by a hive mind that was deliberately holding them back.