Why does the chain tension vary as pedals are turned?
Possible causes:


There is "play" in the bottom bracket bearings, this could also explain the clicks. Usually this is quite noticeable, and you can check it by grabbing the crank-arm and trying to move it sideways. Usually this is not the cause for variable chain tension on singles;
The chainring is "eccentric", either because of haveing been tightened off-center, being manufactured off-center (some defect or poor quality), or damaged. The best way to check is to suspend the bike or turn it upside-down, then slowly spin the cranks do see if it looks off-center. If it is badly assembled, you can try to loosen the bolts and recenter it manually;
The sprocket might be off-center. This can be very subtle, but you can see the difference between the sprocket or the chainring being off-center depending on how many times per turn the chain reaches its tight spot, since the sprocket has a much smaller diameter;
The chain itself can be unevenly worn out, or be uneven from factory. This can happen with very cheap chains, and can cause even a combination of slight unevenness of chain, sprocket and chainring to create a somewhat "random" pattern of tightening and loosening of the chain as the bike rolls.


As for the consequences, I would say that the main problem would be to adjust the rear axle too tight, as this would cause excessive mechanical stress on the drivetrain, or to assemble it so loose that it falls off. Except if you ride fixed or coaster brake (when the lag while pedalling backwards would become a bit annoying), there is not much problem in letting it a bit more on the loose side (enough not to reach an overtight position).

Also, if you use good quality chains and replace the chain often, the rest of the drivetrain tends to "age well" while wearing, because the teeth tend to "even out" in the process. But if you use crappy chains, even good quality gears will suffer a lot more...

Hope this helps!