I took an online English lesson this morning. I took an online English lesson this morning.

The original sentence is correct, but another option is "I had an online English lesson this morning."

I’m shy, so speaking fluent English is still difficult for me. I’m shy, so speaking fluent English is still difficult for me.

I’m shy, so speaking fluent English is still difficult for me. I’m shy, so speaking fluent English is still difficult for me.

You can put a comma here because if you split this sentence into two (taking out the transition word "so") each part works as an individual sentence. e.g. : "I'm shy. Speaking fluent English is still difficult for me."

These lessons changed my old way of thinking that I must not make a mistakes. These lessons changed my old way of thinking that I must not make mistakes.

1) using "old" here emphasizes that your thinking changed :) 2) "make a mistake" also *works*, but this sounds better! e.g "don't drive cars on this road" instead of "don't drive a car on this road"

These lessons changed my way of thinking that I must nocouldn't make a mistakes. These lessons changed my way of thinking that I couldn't make mistakes.

Your sentence isn't wrong! I chose the word "couldn't" here because it implies a past tense which reflects your past way of thinking, which you don't think anymore (i.e., before you felt like you *couldn't* make mistakes, but now you feel like you *can*). However it does feel slightly more informal than "must not", so it really depends in your writing style! I also changed "make a mistake" into "make mistakes". Even though this isn't incorrect, I think it's more typical to use the singular ("make a mistake") when discussing a specific incident. For example, if you were talking about a typo, or the mispronunciation of a word.