I didn’t expect two things after reading your messages: one is that the net engineer needs to take a plane to go where you are to check the problems with your wireless1. I was surprised by two things after reading your messages: one is that the net engineer needs to take a plane to go where you are to check the problems with your wireless network.¶
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2. I was surprised by two points in your messages: one is that the net engineer needs to take a plane to go where you are to check the problems with your wireless network.¶
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3. Two details of your message surprised me: ...¶
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4. I was surprised to learn that an engineer needs to fly to your location to check the network.

1. This is a minimal correction. The key here is that the use of 'expect' is in conflict with an event that has already happened/transpired. You 'expect' something to happen, but once it happens, you can no longer 'expect' anything because it has already happened.

2. This is a more significant modification. Here we are emphasizing how the content is 'within' the messages. It is more direct/succinct and easier to understand.

3. Another way to phrase this.

4. This is a substantial modification. There is some redundancy that you can eliminate. For example, you needn't mention it is a 'net engineer' if the engineer will be repairing a wireless network. And to 'take a plane to go where you are' can be more naturally stated as: 'fly to'.

It is really unbelievable.

This sentence is absolutely correct. But here are some thoughts: the word 'unbelievable' can certainly be used here, but it is a bit strong. You'd have to believe here because it was just told to you (unless the person is lying). The word 'incredible' might be more natural. Here are a bunch of alternative ways:

That's really incredible.

Quite surprising.

Shocking.

Because we usually always communicate in English, you have hardly written or spoken Chinese, so that these sentences exceed my expectations.

This sentence is completely correct, but it a bit stiff/formal/verbose. Below are a few ways to improve it:

1. As we usually communicate in English, you've hardly written or spoken Chinese, so your sentences exceeded my expectations.

2. We usually communicate in English, not Chinese, so those sentences exceeded my expectations.

In one of those videos I heard a crow chirping, exactly the same as I heard when I was a kid.

This would be better broken up in two sentences.

In one of those videos, I heard a crow chirping. It's exactly the same as I've heard as a kid.

Improvements to the second sentence would be:

- I would hear identical chirping as a child.
- I heard identical crow chirping as a child.

'Kid' is a bit more colloquial. I believe the difference is similar to 小子 versus 孩子。

They are planted along some roadsides in our city, and a. Among them, Chinese roses are too. They are our city flower.

The last clause 'Chinese roses are our city flower' is not correct. It stands disconnected from the previous ideas, so you'd need a bit more explanatory information or a separate sentence.