I have been writing Korean for quite some time now, since when I was about 5 years old. I have always practiced writing and practiced some typing, but it's hard to remember which key is which character because I'm typing on an English keyboard. I have come a long way and have seen many patterns of words, clauses, and sentences. I noticed certain characters being added to words in specific cases such as time of action, possession of an object, and towards a direction. Simple and complex sentences are very different in their syntax because simple sentences don't contain these additional characters that would be implemented in complex sentences. I think that understanding the circumstances of which you are communicating in helps you know how to effectively and appropriately communicate.
A couple writing exercises that we've done is using different conditions where I need to write to someone, practice writing answers to prompts, and journaling.
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Yes, I have started writing in my target language, that being Korean. I started writing it when I took a Korean class while studying abroad at Yonsei University last spring. In short, I started writing (and learning) Korean a little more than a year ago. Personally, I prefer to write in Korean by hand, as that way it is easier to remember all the spelling and clauses. Although, typing in Korean is also pretty good, especially on a phone, since it can autocorrect and complete words for you, as opposed to traditional writing methods. Conversely, typing Korean on my laptop is really difficult since I can only see the English letters on my keyboard, and writing words is a bit of a guessing game.
I have noticed some patterns while writing in Korean that are pretty obvious, such as the structure of sentences (being subject -> object -> verb) and how all verbs end exactly the same (depending on the tense). Simple sentences are different to complex sentences as they do not contain words and/or syllables that are specifically designed to connect sentences together, pretty similar to English.
One short writing exercise for my target language that I think is pretty helpful is to write commonly used syllables, so it becomes natural to write them in commonly used words. Another exercise is to write all forms of a verb like "to eat" or "to go" that I want to nail down, including progressive, past, future, etc.