Questions about example sentences with, and the definition and usage of "Workday"

A: 今日はこれで以上です

A: @gin58
Our modern workday today is much easier than before.

A: "an employee" is talking GENERALLY about any employee
"the employee" is talking SPECIFICALLY about one employee

A: " Today is my work day." is usually what you would say.

A: × Finally, the last workday of the year today!
✓ Finally, the last workday of the year!

× I'm looking back on 2021 now, then I've got a small success on my new job and I just adjusted to my new team members through this year.
✓ Looking back on 2021, I've had a small success at my new job and I adjusted to my new team members through the year.

× And mostly, I started to live together with my partner.
✓ But mostly, I started to living with my partner.

× Of course, we had so many arguments and discuss the matter to get a good relationship more than yesterday, then we were able to get a warm and comfortable life for each other.
✓ Of course, we have had many arguments, yet discuss the disagreements to have a better relationship than yesterday. Then we will be able to have a warm and comfortable life with each other.

All English speakers would understand what you wrote. Corrections are to help you become more proficient in English.
"Finally, the last workday of the year!"
"Today" is assumed by the "finally" in the sentence.
"I'm" is assumed because you are writing about your experience in 2021.
Work on using "have* and "got" can improve your English.
If you have any questions about the changes I have made, let me know. Enjoy!

A: × After a exhaustive workday on Monday, I have an English lesson.
✓ After a exhausting workday on Monday, I had an English lesson.

A: It means a day in which someone has to work for the entire day. They are saying that the developers of Siri had to work for entire days to keep up with the amount of people who were using it.

A: instead of "in" you should say "with a glass of wine"

A: I cannot study a lot because during the workday I don't have time.