Red: Incorrect

Green: Corrected 

Pink: Suggested for understanding

Aqua: Punctuation correction

About Today 4

Good evening.

Today I talked with my friend of Chinese Chinese friend on Skype.
Even though a little trable trouble occurred, we were able to talk and hear (understand?) everything each other, so
I had a very good time. I want to talk with her again soon. 

Recently It has been cold since three days ago.
It seems that the temperature of recent is as cold as has been cold because it's March.
Spring, please hurry up and come get here!

Thank you for reding reading.

Good evening.

Today I talked with my Chinese friend on Skype.

Even though a little trouble occured, we were able to speak and understand each other, so I had a very good time. I want to talk with her again soon.

It has been cold since three days ago.

It seems that the temperature has been cold because it's March.

Spring, please hurry up and get here!

Thank you for reading.

About Today 4

Good evening.

Today I talked with my friend of Chinese on Skype.

Today I chatted with my Chinese friend on Skype.
Though a little trable occurred ,because we were able to talk and hear everything ,

Although we experienced a small technical problem (say this if you had a problem with Skype), all in all we were able to chat and hear each other quite well.
I had a very good time. I want to talk with her again next time.
I had a very good time. I look forward to chatting with her again.

Recently it has been cold since three days ago.

It has been cold for three days now.
It seems that the temperature of recent is as cold as it's March.

The weather we have had recently seems as cold as if it is still March.

Spring,please hurry up and come here!

Spring, please tarry not (this is rather poetic and old-fashioned...) and be here soon!

Thank you for reding.

Thank you for reading.

About Today 4

Good evening.

Today I talked with my Chinese friend on Skype.
A) Though a little trouble occurred, because we were able to talk and hear everything. I had a very good time and I want to talk with her again sometime. 'sometime' is a much better word for this as the time hasn't been set, while 'next time' implies that it has.

( (A) A little trouble occured because you could talk and hear everything, or, (B) you had a very good time because you were able to talk?)

B) Though a little trouble occurred, I had a very good time because we were able to talk and hear everything. I want to talk with her again sometime.

Recently it has been cold since three days ago.

It has been cold for three days recently. (You are right, but this way sounds more natural to native speakers)
It seems that the temperature of recent recent temperature is as cold as it's currently March.
Spring, please hurry up and come here!

Thank you for reading.