Population growth and death are deterministic. There is nothing random about it. It is possible to design a population similator that exactly matches the growth and death of your population (given the current morale, which is also deterministic and can be simulated). This is true for all colonist types. The only "random" elements can come from how external factors affect morale or population level. Such as if you're having a battle with someone and killing each other's buildings. Also, for morale to have an effect, it must change by an entire level, not just morale 99 to morale 98 or something small like that. It must cross a level boundary to change how it affects population growth or death.
Now I'm going to assume you're not losing people directly from buildings being destroyed while they are still active and have people in them. In which case, morale is the only variable issue. So if you're doing everything else right, then pay attention to your morale.
More nurseries doesn't really help. You need at least one to have children be born. Also, keep in mind there is a hidden progress to "building children", sort of like how factories work. If the building ever becomes inactive, progress stops. Now, each nursery you have also decreases the infant mortality rate by a little, but only as much as an unupgraded medical center would. Keep in mind that medical centers also decrease the mortality rate of workers and scientists, and they take just as many colonists to staff as the nursery does. In other words, it's sort of wasteful to build more than one nursery. You should build more medical centers instead. The only reason for having more than one nursery, is in case one goes offline, such as when scientists die. This is usually only an issue if you're the sort of person that doesn't notice this sort of thing and let's it stay offline for a long time.
The more medical centers you have, the less of an effect each additional one has. They do however keep decreasing the death rate, even if you're well below 100% usage.
There is no way to increase your birth rate other than keeping morale high. Having a large population also helps birth rate, but you can't just "increase" the number of colonists you have and thereby increase the birth rate. But keep in mind that workers contribute twice as much as scientists do to the birth rate, so if you train too many scientists early on, you'll hurt your long term population growth. If the number of colonists is an issue, try to hold off on training scientists for a while. Especially if your population is already low, since then you'd be decreasing an already low birth rate.
As for morale, pay attention to the main "core" morale factors such as building usage. Just check the morale report (ctrl+M) to see those factors and try to keep them all satisfied. These give you a base morale value, and in the absence of events, will determine your overall morale (multiplied by some factor that depends on the difficulty setting and colony your playing (plymouth has a bonus), but is somewhere around 0.8 to 1.2). Once that is taken care of, then you can also start thinking about how event affect morale. Events will either add or subtract from the base value, and the event portion of your morale decays to 0 over time, decreasing in absolute value by 1/2 each period. Things like completing research will give you a positive bonus, as will killing an enemy unit. Losing a building, or a disaster occuring, whether or not the disaster hits you will decrease morale. Even killing an enemy building can decrease your morale. So if you're morale is low, it's a VERY bad idea to go killing residences or agridomes, or other similar non combat significant buildings. Kill those buildings is probably the biggest negative impact your morale can suffer, and if you kill enough of them, it can take quite a while for your morale to recover, during which time your population will probably drop significantly, unless it's already huge (at least 100+ to 200+ people) and has a naturally high birth rate.
And would people finally get over the "clicking cheat" nonsense! It's completely false. It does nothing. At best the status display just updates when you click giving the appearance something is building faster, since normally it only updates every so often and makes the progress appear to lurch forward. We've looked into this just to prove how silly it is. We know where in memory to find the current progress at factories, and we've monitored it, and tried clicking repeatedly. We've done many trials just to be absolutely sure. There is absolutely no hidden affects on the progress. It only gets updated during the expected point during the game loop, and the update can not be affecting by clicking. Further, if this did work, it would desyncrhonize the game in multiplayer. There is no command packet that gets sent when you simply click on a building to view it's status, so if it had any effect, multiplayer games would get out of sync, and probably eventually crash. There are no hidden packets sent either. We can monitor this, and even if there was, we already know what all the packets do, and none of them do anything that this "cheat" would require them to do to work. The memory monitoring made use of hardware breakpoints, so there was no way the value could have been modified without being caught by the debugger, and we've searched for other possible code that could have modified that value had it run, even though it didn't run during any of the tests since the hardware breakpoint caught nothing, and we came up with no likely suspects.
Edit: Forgot a test. Two factories had command packets issued during the same game cycle to produce identical units. One factory was clicked on repeadly, the other wasn't. Both vehicles appeared at EXACTLY the same time. As in, the same game cycle, not just appeared at the same time by someone eyeballing the test.
In short, there is no cheat. If there was any such clicking "cheat", it would have to have been hidden VERY well, because any simple oversight or bug would easily have been caught by the above efforts. In which case if it was hidden, and placed there on purpose, it'd likely be much harder to activate than simply clicking repeatedly, since we'd have caught that by now. Further, there is no room to hide anything in the command packets, since we know how they work. In which case, there is no possiblity of it working in multiplayer. If it worked in any game mode and had been activated during the tests, it would have been caught by the hardware breakpoint. Had it not been activate during the tests, and not caught by the hardware breakpoint, then it's more complicated than simply clicking repeatedly. In which case, people probably aren't activating it (if anything is even there). And even if there was still something there that somehow shortened build time, we probably would have at least found something in the code analysis that would suggest a possiblity of some kind of cheat. So until someone finds some evidence of a not just clicking repeatedly activated progress cheat that evades hardward breakpoints, was specifically designed to only work in single player, doesn't use any of the usual offsets anywhere in code, was never activated during any of the testing trials, (but is somehow still easily activated by people playing?) then I'll maintain that any such claims as to the existence or validity of this "cheat" to be about as believable as people claiming they were abducted by aliens and given an anal probe.
