If I change the order and say "article about your work in Business Weekly", does it sound weird?

Can I use "as the fact that" instead of "for the fact that"?

No, "as the fact that" can not replace "for the fact that".

What does "get that out" mean here?

To "get that out" in this context means to "get the words out" or to be able to verbalize or say what you are trying to say. Ex: I have an idea in my head but I can't get it out. Ex: It's hard for him to get out what he wants to say.

In a resume, what does "job description" mean? Also, what should I write on the job description?"

Actually, a "job description" is usually written by the employers! It explains the tasks, functions, and responsibilities of a job role or position and how it relates to the company. What they're saying here is that employers usually have keywords, such as skills or experience that they include in the job description and that they expect good applicants to mention in their CV or resume. Ex: I really liked the job description for that company. I think I'll apply there. Ex: My career counselor always said to have keywords from the job description inside my resume.

The dictionary says "turtledove" is a type of pigeon. Then what does she mean by "my turtledove"?

You are correct! The turtledove is a type of pigeon. However, it is a bird species known for having strong, affectionate bonds with its mates. It has been a symbol of love and devotion in literature for many years, which is why "my turtledove" was also used in the past as a pet name meaning "sweetheart" or "beloved mate.” Nowadays, this name has become a bit outdated and isn't really used anymore.

Is it okay to replace "this is" with "he is"?

Yes, you can replace "this is" with "he is." The speaker has used "this is" to add a feeling of emphasis and formality, but you can use "he is" to express the same meaning. Ex: This is our team's ace. Ex: He is our team's ace.