Why does snow stay after a snowfall?

Just as a complement to Ziggurat's answer: you can try to estimate the time required for the sun to melt a certain quantity of snow by yourself.

• The energy required to melt a mass $m$ of snow is $$Q=L m$$ where $L$ is the latent heat of fusion. For ice, $L=334$ kJ/kg.

• The density of snow $\rho$ ranges from $100$ to $800$ kg/m$^3$

• Solar irradiance $I$ ranges from $150$ to $300$ W/m$^2$.
• The albedo of snow (percentage of reflected sunlight) $A$ ranges from $0.2$ for dirty snow to $0.9$ for freshly fallen snow.

If the surface exposed to sunlight is $S$, the absorbed energy in the time interval $\Delta t$ will be

$$E_{in}=(1-A) IS \Delta t$$

If $V$ is the snow volume, the energy required to melt it will be

$$E_{melt} =L \rho V$$

Equating these two expressions we get

$$\Delta t = \frac{L \rho V}{(1-A)IS}$$

Assuming $A=0.9$, $\rho=300$ kg/m$^3$ and $I=200$ W/m$^2$, we get, for a sheet of snow of surface $1$ m$^2$ and thickness $1$ cm, $\Delta t \simeq 5 \cdot 10^4$ s, i.e. $\simeq 14$ hours.

This is a very rough estimate that doesn't consider conduction processes. But anyway, you can see that even if we assume a pretty high irradiance we need a considerably long time to melt a modest quantity of snow. If the snow is in the shade, the value of $I$ will be less. Also, for snowmen, since we would be talking about compressed snow, the value of $\rho$ could be $2-2.5$ times larger.

For one thing, snow has a high albedo (it's very reflective) so it won't absorb much sunlight and warm up through that process. Thus, it will have to mainly heat up from convection, which isn't terribly efficient. Snow is a good insulator, so only the surface will be prone to melting. Also, the heat of fusion must be overcome in order to achieve the phase change. This answer isn't very well organized, but hopefully it conveys that there are many factors that work against the melting of snow.

There is some heat energy needed to melt snow, and it corresponds roughly to the energy needed to heat water by 80 degrees Celsius - this is quite a lot even compared to other substances, and so it takes few days of sunny weather for all snow to absorb the required energy from its surroundings.

Few notes to your sub-questions: 1) Melting the same mass of ice and snow requires the same energy, they do not differ chemically. Ice however absorbs more sunlight. 2) Yes, the air above snow in windless weather is colder and helps to insulate the snow from the warmer environment. This is particularly true in trenches and valleys. 3) No, snow usually does not contain any salt, but if it did, it would melt faster.