Why is it that raindrops don't collide and 'stick together' on their descent to Earth, arriving in streams rather than separate drops?
Have a look at the Wikipedia article on raindrop formation. You'll also find lots of articles on raindrop formation and growth by Googling raindrop formation or something like that.
Raindrops do coalesce, but they also fragment, and the eventual size is a balance of the two processes. The fragmentation occurs because of the forces from turbulent air flow. Turbulence can cause droplets to collide, in which case they may coalesce, however it can also break apart large droplets.
Incidentally, a stream of water is unstable at low flow rates because of the Plateau-Rayleigh instability so it's very unlikely you could get a continuous stream of rain even under ideal atmospheric conditions. The closest you would get is a series of droplets in a line. However in the real world even the slightest turbulence would scatter the droplets and lead to the random distribution of droplets that we see.
Water starts to fall from the clouds when the drop size reaches a critical point which depends on a lot of factors such as the strength of upward currents, but also air density and gravity acceleration. When the drops leave the cloud no more collisions take place and their size is fixed.
To create a water stream we need to concentrate water from a large area to a local point. This is done by each roof and gutter, but could this happen in the atmosphere?
Let's think about this: air currents will need to point from different directions to a local spot bringing there water from a large volume. This would require air to compress approaching that spot, which would divert this current. So, even from an intuitive point of view, this is not likely to happen.
First of all streamline is a flow in which every drop at same position has same velocity, this is true for rain as long as wind is neglected!
What you seem to be asking is that why there is significant time lapse between individual drops of rain as opposed to a shower and/or why the flow is like a shower and not that of a tap.
Well the flow of a tap is based on water leaving the container through one opening, as the water particles crowd together at the opening they come out together. In a shower there are multiple apertures for the water to leave and hence the separate streamlines do not always coalesce unless the shower is curved to make them coalesce.
A cloud acts like both in different times, mostly it behaves like a shower and has multiple openings sending separate streams of water, but sometimes it becomes one giant aperture (cloud burst) and all the water it contains comes down like that of a tap/waterfall resulting in flash floods at many times.
As to the othet part, in a normal shower the distance between drops is very less because the travel very less. Lets say the shower is at a height of 5m and a cloud is at a height of 500m, if droplets come one every second from both then lets watch 2 consecutive drops.
For the shower, when first drop hits the floor, the other drop is still in the shower but only at a distance of 5m, as rate increases the distance becomes negligible.
But for a cloud, when first drop hits the ground, the next drop is 95m above the ground, even after rate becomes equivalent to that of a normal shower there is noticeable difference and hence you observe drops than streams!