Since I cannot type simplified characters, I typed with traditional ones. Please bear with me and use a dictionary if you have any characters you don't recognize.

不管我去台湾，我会跟我的比利时的朋友多联系。并且我会请他门去看我。

This sentence is grammatically incorrect, and it can be modified in several ways.

(1) 不管我去不去台湾，我還是会跟我的比利时的朋友多联系。

　　　不管 is basically equivalent to "whether" in English, and just like english a negation tag "or not" is usually needed when the condition is stated specifically.

　　　還是 means "still" here. It sounds more natural to have it after the sentence led by 不管.

(2) 就算我去了台湾，我還是会跟我的比利时的朋友多联系。

　　　就算 means "even if" or "even though". The sentence sounds much natural to use 就算, as Taiwan and Belgium are two very far countries.

(3) 我 比利时的朋友 or 我的比利时 朋友

　　　just a minor fix, but in Chinese it is mostly preferred to avoid using two 的s in one noun phrases.

(4) 并且我会请他门來看我。

　　　It makes more sense to ask your friends to "come" instead of "go" to visit you. The direction verb in Chinese languages is usually 來 when mentioning the first person (I, we).

　　　並且 is quite formal, so it would be better to use other words here, such as 而且、還會...

好好想吧，否则你会后悔！(correct)

由于他不听医生的话，到现在他的病都还没好了。

The usage of 由於 is correct, although 由於 is a bit formal and objective and 因為 would be preferred for describing subjective or personal matter.

The second sentence is very unnatural and should be changed to the one above. 還沒 and 了 usually don't coexist, as they are both aspect markers, and that an action cannot be "not yet done" and "completed" at the same time.

我们写完作业了还有时间了，于是也順便预习了明天的课。

The usage of 於是 is correct, although 於是 is a bit formal and objective and 所以 would be preferred for describing subjective or personal matter.

完 is an aspect marker that should go right after the verb. 還有 and 了 usually don't coexist, as an action cannot be "still progressing" and "completed" at the same time.

順便 is added to make the sentence sound more natural.

The 了 in the second sentence should be placed right after the verb to indicate completion. If 了 is placed at the end of the sentence, it would mean that the whole action have an impact to action afterwards or current time, which sounds weird since no other context is provided.

即使他们离婚了，他们也还是好朋友。(correct, although adding an 也 is more natural)

麻烦你们帮我看看这样写对不对？

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既然你知道你犯了个错误，为什么还没给他道歉？(it's right)

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不管我去不去台湾，我会跟我的比利时的朋友多联系。并且我会邀请他们来看我。(When you use "不管", you need to list multiple chooses or use a question, for example:

不管他是贫困、患病、或者残疾，甚至死亡(four chooses)，你都会忠于他吗？(wedding oath)

不管有没有希望(two chooses)，我都会尝试。

不管发生什么事情(A question, also you can think it as infinite chooses)，我们都会在一起。

you can think it as "whether" or "no matter".

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好好想想吧，否则你会后悔！(You use "否则" rightly, but have a mistake in sentence.)

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由于他不听医生的话，现在他的病还没好了。(The same as above)

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我们写完了作业还有时间了，于是也预习了明天的课了。(I think you understand "于是", but you have some mistakes in grammar, so the sentence is not smooth.)

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即使他们离婚了，他们还是好朋友。(This sentence is no problem, but I think it runs counter to reality:(   )

麻烦你们帮我看看这样写对不对？

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不管我去台湾，我会跟我的比利时的朋友多联系。并且我会请他门去看我。

（因为去台湾和跟比利时的朋友联络没有关系，所以不可以用，如果是我的话我会写：不管我去不去台湾，我都会和我的台湾朋友联系，还有应该要打我的比利时朋友or我比利时的朋友，虽然这样打也可以理解，但是会有点奇怪。）

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好好想吧，否则你会后悔！

（好好想想吧，否则你会后悔）

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由于他不听医生的话，现在他还没病好了。

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我们写作业完了还有时间了，于是也预习明天的课了。

（我觉得前面加一句“因为”会比较顺）

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即使他们离婚了，他们还是好朋友。

（这句对了）

麻烦你们帮我看看这样写对不对？

《》里面只能写书名，诗名，歌曲名，舞蹈名，画的名字，作文题目，文章题目，影视作品名，动画片，电视剧名等文化或文艺类型的题目。例如《三国演义》。

如果想要强调一个字、一个词、一句话，你可以用“”。例如“技巧”，“既然”。

一个人所说的话要用“”。例如，我对妈妈说：“我爱你！”。

既然(√)你知道你犯了个错误，为什么还没给他道歉？