I woke up at 6 AM this morning actually, but I'm actually not sleepy now, because I always wake up at 5 AM every day. I woke up at 6 AM this morning, but I'm actually not sleepy because I wake up at 5 AM every day.

Even though your sentence is clear, it's more natural to say either "I always wake up at 5 AM" or "I wake up at 5 AM everyday" instead of using both. Another way to say it would be: "I usually wake up at 5AM".

I actually woke up at 6 AMam this morning actually, but I'm not sleepy now, because I alwaysnormally wake up at 5 AMam every day. I actually woke up at 6am this morning but I'm not sleepy now, because I normally wake up at 5am every day.

I've been studying English for months so waking up early morning is regular for meI regularly wake up in the early morning. I've been studying English for months so I regularly wake up in the early morning.

It'd say "I regularly" rather than "is regular for me". It's also possible to say "waking up early in the morning" but what you have is correct too.

I've been studying English for months, so waking up early morning is regularhas become a habit for me. I've been studying English for months, so waking up early has become a habit for me.

And I prefer the morning to the night. And I prefer the morning to the night.

Could also say, "Plus, I prefer the morning to the night." since this is an extra add-on sentence at the end.