I will comment from the beginning:

I still have no clear idea of your temperature classification but I'm going to take a guess:
extreme is below -38, very cold is below -10, cold is below 0, cool 0-10, mild 10-18, warm 18-22 ?, hot 22-32, very hot is over 32 degree Celsius


BSh and BSk: small detail but it's S and not s, otherwise it mean summer dry
these two climates can have a rainy season, some places could receive up to around 200mm of rain in a month because they are subject to the monsoon (mostly the hot one). Large parts of the steppes in Africa and Eastern Asia are affected. Examples include Niamey and Hohhot.

BSk : is usually found at higher altitude than the hot desert

Dont forget that : (I know it's pretty complicated!)

B. Dry climate: Annual evaporation is greater than precipitation (also called potential evotraspiration). To determine whether a location has an arid climate, the precipitation threshold must first be determined. The threshold value (in millimetres) is determined as:
• If less than 30% of annual precipitation occurs in the summer : Annual precipitation (mm) < 20 × average annual temperature (°C)
• If more than 70 % of annual precipitation occurs in the summer: Annual precipitation (mm) < 20 × average annual temperature + 280
• Else : Annual precipitation (mm) < 20 × average annual temperature + 140
o If annual precipitation is < 50 % of the threshold = BW: desert climate
o If annual precipitation is between 50 and 100 % = BS: steppe climate

So the steppe climates can be more or less rainy if the evaporation is very high too. The same principle stand for colder climates too. They receive much less rain than hot climates but are still considered wet because the evaporation is much lower.

(detail) Also, according to the classification, it's possible to have cold winters in BSh as long as the average yearly temp are high enough. Rare but possible

(detail) BWn: it's an additional class for desert near water that have a small temperature variation like Namibia. I was considering incorporating it


Csb: monthly temperature never go over 22, so it's not hot and not cold either because it never go below 0 (monthly)
I just like to specify that Csc form usually at higher altitudes at it's not very common

Cwb/Cwc: monthly temperature never go over 22, so it's not hot and it's even colder for Cwc, they are located at higher altitudes

Cfb: it pretty sure that the temperature rarely go under 0. And what do you mean by very wet ?

Cfc: : it's hard to say because only a few places are said to have this climate. Like Reykjavik: the temperature difference are small and winters are cool around 0 degree.

D climates: if I have the temperature right for the extreme cold, it should only be applied with : Dfd, Dwd, Dsd and the other colder climates. I tried to find other climates in the D group with winter months under -38 but I could not.

Dfa,Dfb: you inversed the summer and winter precipitations.