Is there a lake whose sole source of water is rain and has a hole at the bottom draining it?
I think there is some misunderstanding of the water balance of a lake, or to put it differently, why some holes in the ground are dry while some are filled with water (known as lakes). The linked articles contain lists of closed lakes, but I'll also try to explain how lakes are formed. 

Fresh water comes from the rainfall (and snow and moisture) that falls on the ground, and if it does not evaporate, it will eventually seep down into the groundwater when it reaches the groundwater table. 

The groundwater flows towards lower elevation, known as hydrostatic head, and eventually, it reaches the sea level. If you would dig a number of holes from the sea towards the mountains, you'll see that if the holes are deep enough to reach groundwater (wells), the elevation of the groundwater surface will form a slope. If the geology is such that the water only flows slowly through the ground and the precipitation is high, the slope can be steep, but if you have e.g. a sandy ground and low precipitation, the groundwater table is subhorizontal. Indeed, in areas with high evaporation, the slope can face away from the ocean, towards a local depression as e.g. the Dead Sea or Death Valley.

If the groundwater table crosses the ground so that the elevation of ground is lower than the hydrostatic pressure of the groundwater, the groundwater will be exposed and form a lake or enter a river. The lakes and rivers are manifestations of the groundwater level. Sometimes there is a delayed response time as water flows much quicker in a river than in the ground. This causes flash floods to flow above the ground if there is not enough time for surface water to penetrate the surface. Like a coffee filter overflooding if one pour water in it too fast. Given enough time, the surface water will enter the groundwater and flow towards the ocean or evaporate. 

Sometimes there are features that speed up the flow, e.g. karsts or gravel lenses that have a very high hydraulic conductivity so that water can flow rather fast through. Interesting phenomena can occur that control this kind of quick drains if e.g. the drainage gets frozen and clogged in winter or there are some other seasonal variations. This is maybe what you describe as a drain hole, but usually, lakes without outlets just adjust the water back into the groundwater by penetration and by evaporation of the surface. 

Dissolved salt is concentrated in lakes without an outlet and they are called closed lakes or endorheic basin. If the climate is changing, an open lake can become a closed lake and vice versa. 

A lake is always a temporary feature. They fill up with sediments over time and becomes a part of the river path. As you noticed, there is always a special reason for a lake to form, e.g. rapid post-glacial rebound, rifting or an impact crater. 

Ponds and tarns without a visible outlet are rather common, at least in post-glacial areas as Canada or Scandinavia, but if a lake has enough inflow, its level will rise until it gets a visible outlet. Depressions without a visible outlet are called sinks.

How pure the water in the lake is depends on a number of factors. Silt and clay come from fast-flowing inlet rivers or wind. The residual time and current for the water in the lake defines how much of the sediments that have time to settle. The bathymetry is also an important factor as local depocenters can take care of most of the sediments. Fine sediments can take a long time to settle. Finally, the biology of the lake is naturally very important.