Surprise!

It has been two years since the last time I wrote you a Japanese birthday card...
Since you are still talking about the presence, 'has' is the correct form to use.

Thank you for always messageding me and shareding with me about your current life, future goals, and even worries about a relationship with me. 
You can either say "always having messaged" or "always messaging". "Always messaging" sounds a bit simpler, and it seems like your friend continues to message you (and share with you about his/her life), so 'messaging' and 'sharing' should be in the present progressive form. I also put the prepositional phrase 'with me' before the list of things your friend shares with you, otherwise the original meaning you convey almost sounds like you friend has relationship problems with you personally.

This postcard iswas bought from a Taiwanese designer, Fifi, working in Paris, Fifi. 
The action of you buying the postcard happened in the past. Thus, 'was bought' is the correct passive tone of describing the action of buying the postcard in the past. Additionally, it would be easier to follow if you place the name Fifi right after the description "Taiwanese designer", instead of after the place (Paris).

I love her artworks so much that I bought a paper tape, a coaster, and another postcard showing the autumn in Taiwan. 
Artwork can already be considered plural. You can also either say "showing autumn", since the seasons don't have an article in front of them, unless they are used as an adjective describe something, such as "the autumn scenery".

They are very moisturizing and sold pretty well in Japan and Korea as well.
Do you mean to say "pretty well" as in the masks are popular in Japan and Korea? If so, you could simply say that "they are moisturizing and popular". Otherwise, 'pretty well' does not make sense in this sentence.

...let the essence come out, and applying it on your skin again. 
You need to specify what you are applying. Since you are talking about the essence, you can simply use the pronoun 'it' to refer to the essence. Additionally, in English, the prepositional phrase for applying things like skin care is "apply __ on".

The postcard is too small so I can just put the words in brief write a brief message.
"Put the words in brief" sounds a little awkward - you can just say that you can only write a brief message.

Next time when you get a chance to come to Asia, remember to drop by my country. I just can't wait to let you experience the cuisine, night markets, and hot spots in Taipei. 

...so don't worry that you will have about having communication problems with my family.

Let me know if you have any questions !