Orc populations certainly suffered in the South, but no where near to the extent that the Red Book claims. While large swathes of Mordor were vacated following the Battle of the Morannon, substantial groups of orcs removed to Khand, others formed communities along the eastern shores of the Lake Nurnen. The author of the Red Book also exaggerated the extent to which Mordor was depopulated. The region never held much appeal, even during the long years when Sauron was dormant in the third age. Recent archaeological evidence has revealed Orc settlements, especially in the foothills of the Ered Lithui, that date to the first decades of the Fourth Age. So while Mordor did experience widespread depopulation, it was not to the extent that propagandists of the early Reunited Kingdom claimed. Furthermore, populations of goblins in the North remained relatively stable throughout the period.

On another note, I have to warn readers not to use the Red Book uncritically. It's certainly a wonderful resource, but it is still a product of it's time and the biases and exaggerations of the author are evident throughout. The text is also interpolated in several places, and while it is likely that Frodo, or at least his notes, we're integrally involved in the composition, even Tolkien recognizes later Gondorian intervention in it's textual history.

I'm on mobile at the moment and am more than glad to expand on this question when I have access to a computer

EDIT:

To expand a little on my previous post: As /u/TEmpTom points out, the reoccupation of Moria was a bit of a blow to the goblin communities of the Misty Mountains, but incidences of large-scale violence were relatively isolated. There appears to have been goblin settlements in Moria throughout the Third Age, but many historians have theorized that when Balin and his party arrived, their population had dwindled, likely owing in part to the wakening of the Balrog. By the end of the War of the Ring, most goblin populations had removed further to the North, and the group that chased the Fellowship through the Mines of Moria was likely smaller than what the Red Book describes. Needless to say, over the course of the Fourth Age, goblin settlements in the northern reaches of the Misty Mountains around the old kingdom of Gundobad increased, as did those in the Grey Mountains. The traditional Gondorian narrative of mass killings is a bit overzealous, most orc populations simply migrated and resettled in areas not directly under control of the allied forces.

My focus is on goblin and orcish society, so I can't speak as much on the early history of the Reunited Kingdom of Arnor and Gondor, but populations in the Upper Anduin slowly increased. We see this in the archaeological record as well as in a series of letters between the Kings of Rohan and their kin that still dwelt around the head waters of the great river. The region had been generally depopulated over the course of the Third Age, a decline hastened by the instability of the wars that brought the Third Age to a close, but settlement initiatives among the Rohirrim following the waning of the expanded realm of Lothlorien and a general population boom following the War of the Ring saw the removal of large family groups to the north, searching for new opportunities afforded by increased regional stability and available tracts of fertile land.

Finally, a question on this sub I can actually answer. To put it bluntly, the Battle of Morannon itself was rather insignificant, however it's only real purpose and consequence was to facilitate the destruction of the One Ring. The real effect on the 4th Age was ultimately the death of Sauron, and subsequently his Empire. Due to the centralized command and leadership structure of Mordor's empire, lacking any central leadership, Sauron's armies and vassals scattered and declared their independence.

The Orcs, being originally bred as a servant/slave warrior caste were the first to desert Mordor, disintegrating into tribal communities and small warbands still common place throughout the Misty Mountains, Grey Mountains, and parts of Ithilien for over a century into the Fourth Age, occasionally they raided the territories of the human territories, however they never really posed a unified threat after Sauron's death. During Aragorn Elessar I reign, he ordered a systematic genocide of the Orcs within Gondor's territory, and when the Durin VII reclaimed Khazad Dum, the Orcs became very few in number.

Sauron's Empire served as a Axis power to the influences of human kingdoms in the West, most notably, Gondor, Rohan, and the Rhovanian remnants in the North. When it collapsed, Elessar I was able to conquer many of the old territories taken by Sauron's vassals thousands of years ago. The port city of Umbar was the first of these reconquests. After the Corsair fleet was destroyed during its attempted assistance of Mordor during the Siege of Minath Tirith, Elessar was able to send the Gondorian navy to recapture the city, and it's surrounding provinces. From Umbar, Elessar launched and invasion of Near Harad, and subjugated most of the region before leaving it as a tributary state to the Reunited Kingdom after realizing that it was too large to rule directly. After some years, Elessar attacked the Wainrider and Balchoth tribes, and reclaimed a large portion of western Rhun, forcing the Easterlings to migrate further east. Of the human vassals of Sauron, only the kingdoms of Khand, and the tribes of Hildroen and Far Harad evaded Elessar's brutal reconquests due to their distance from Gondor.

Elessar's most notable contribution after the War of the Ring was the reunification of the Dunedain kingdoms of Gondor and the remnants of Arnor. His massive campaigns of reconquests would not have been possible without support and contributions of the Northmen and the Dwarves. His alliance with the Dwarves also supplied his armies with the metals, weapons and armor. The King of the Lonely Mountain was very happy give out very low interest loans to Gondor's campaigns.