According to Laws and Customs of Eldar the elves's average number of children is four (rarely more, the absolute record is seven, sons of Feanor) and between one child and another usually there is gap of up to hundred years (so they could focus on rising one child at a time with exception of twins :). Also about their sexuality, ,,union of love is indeed great delight and joy for them" so it's not a problem of their sex drives but rather other natural ways of their lives (the greater part of strength is required for elves to bear children).
Their numbers though is hard to estimate, during the 17 years from Bilbo's party to War of the Ring many High Elves left Middle Earth (and most of those who stayed remained living in Rivendell, Lindon and Grey Havens respectively to the Fourth Age but no longer could sustain large armies, probably no more than to protect themselves at need), we know more about Wood Elves of Mirkwood, their number was in thousands (during the Hobbit Elvenking Thranduil took only one thousand warriors because he didn't expect battle and sought only to secure the Mountain for it's treasure, larger force must have stayed at the Woodland Realm for protection of it's borders), the same with Lothlorien they succesfully defended against Dol Guldur attacks and later led invasion on this fortress (and settled in southern part of Mirkwood calling it East Lorien, also in Lothlorien wood Caras Galadhon was the ,,chief dwelling" implying more than one). There are also references to other elvish clans of Avari living somewhere in the east (six clans listed by Tolkien by their names only).
As for other races, men were increasing in numbers and only suffered losses during wars, plagues that could wipe out whole communities and natural disasters, like great flood that destroyed city of Tharbad.
Though Eriador was struck with such a plague and war with Angmar highly depopulated the region, many wide expense of lands were empty, even in late Third Age there were still scattered inhabitants. Besides such places as Bree the race of men lived also in lands of Minhiriath (,,a few secretive hunter-folk") and Enedwaith (part of it that lies near mountains, in Dunland and on coastlands ,,numerous but barbarous fisher-folk" especially in woods of Eryn Vorn) and somewhere in the wild are hidden settlements of Dunedain (some sources tell they live in the Angle south of Trollshaws).
Hobbits quickly increased in numbers especially in times of peace and plenty (Shire seems highly populated and hobbit families are usually very large, though absolute record in number of children belongs to Samwise Gamgee who had 13).
Rohan well we know only of two named large cities Edoras and Aldburg. There are also other named settlements in Rohan: Grimslade in Westfold (ancestral home of Grimbold), the Deeping Coomb ,,rich valley" near Helm's Deep is also home to many people, the valley of Harrowdale is very densely populated and there are two settlements which names were given: Upbourn and Underharrow. From Edoras and surrounding area Theoden brought to Hornburg about thousand armed men, so population was quite big, near the capital, besides this over one thousand men there must have been women, children, old who could not fight. Mainly Rohirrim lived in the valleys of White Mountains and some assume their number as a nation must have been below 100 000 (Wikipedia article estimates that total number of the Rohan's army was about 20 000 and usually soldiers numbers are exceeded by non-combatants, Gondor army is speculated to number total 30 000 men containing all forces from the coastlands and fiefdoms that sent only small part of their forces to Minas Tirith, Gondor also has lot more cities than Rohan so population of Gondor must be bigger than in Rohan)
Dwarves, there is no source showing how numerous was their race, one can assume they numbered many thousands from the Longbeards clan only (there are Seven Houses, Durin's Folk or Longbeards were most numerous, but the rate of birth is low due to small number of dwarf-women) the dwarves maintained many colonies: Erebor, Iron Hills, Ered Luin (Blue Mountains inhabited by three Houses, Longbeards that came with Thrain after Smaug's attack, Broadbeams and Firebeards that didn't go to Moria after the First Age ending with War of Wrath), temporarily in Dunland, settlements in Grey Mountains (abandoned due to the dragons from Withered Heat), later new colony in White Mountains in the Fourth Age. In the 17 years gap between party and War the ancient Great East Road was said to be filled with ,,dwarves..in unusual numbers" many of whom seems to be refugees from four eastern Houses and many Wandering Companies of elves traveled through Eriador. Hard to say much about their numbers or where their permanent settlements were (for example Gildor Inglorion mentioned dwellings of his people but it isn't sure if he meant Rivendell or some other place).
There are of course many other places and whole nations in the East, South, Rhovanion and in the far north (like Lossoth) in unspecified numbers.