It snowed today and it's the first snow of thise year.

we can say: the first snow this year (or) first snow of the year (it's understood we mean this year)

Before this Chinese New Year, it was said thathe weatherman forecast heavy snow would come, but it didn't happen.

In American English, it's common to avoid passive voice whenever possible.
We use passive voice in 3 situations:

1) When we don't know who did something: The car was stolen (we don't know who stole it)
2) We don't care who did it: The mail was delivered this morning (we don't care who the mailman was)
3) We don't want to admit guilt: Child to mother: The window got broken. The child doesn't want to admit that he broke the window.

In your sentence, we know who spoke (the weatherman), so it's better to avoid passive voice.

The snow today was smalllight and it almostpractically melted just in the afternoon.

The snow willould be as high as my thigh, but in the past few years, it only snowed one or two times and the snow was only as high ascovered my feet.

The snow would be = repeated action in the past

I have been wondering if some day it won't snow one day here at all because the weather has been gradually becoming warmer.

It snowed today and it's the first snow of thise year.

In the past, the first snow would come before the Chinese New Year, but in recent years, the first snow often caomes after the Chinese New Year.

I used to doubt ifdidn't think it would snow during this Chinese New Year.

The snow today was a small amount and it had almost melted just inby the afternoon.

I docan't guarantee it will snow again around thise Chinese New Year.

Do you mean you don't think it will snow again this year? Or do you mean around future Chinese New Years' in general?

In the last decade, where in my placI live, it snowed less frequently, didn't snow too much, and ithe snow melted quickly melted.

Decades ago, when I was young, every year it snowed three to five times per year and it snowed very heavily.

The snow will beas as high as my thigh, but in the past few years, it has only snowed one or two times and the snow was only as high as my feet.

I have been wondering if one day it won't snow one day herehere at all because the weather has been gradually becoming warmer.

I used to doubted if it would snow during this Chinese New Year.

"Used to doubt" implies a more distant past.

In the last decade, here in my place, it snowed less frequently, didn't snow too much, and the snow melted quickly.

I get that "didn't snow too much" is referring to the duration of the snow, when it did snow. It could also describe the frequency though, so I was confused when I first read it and thought you were saying the same thing again. You could emphasise that you're talking about duration by adding something like "and when it did snow, it didn't snow too much" to avoid confusing the reader.

Decades ago, when I was young, it snowed every year it snowed three to five times and it snowed very heavily.

The snow willould be as high as my thigh, b. But in the past few years, it only snowed one or two times and the snow was only as high as my feet.

The "but" is contrasting the height of the snow and the previous sentence. When you have "The snow would be as high as my thigh, but..." the but is connected very strongly to just that sentence, and loses the connnection to the previous sentence.

I have been wondering if it won't snow here one day here because the weather has been gradually becoming warmer.