[Translation Challenge] How Do You Translate “To Be Home Late”?

Nice try! One thing, your sentence would be perfect if you unify the subject of「部長が仕事終わるまでにしたい事が頼まれたし」 part

If you’d like to make 部長 subject: 「部長が（私に）仕事が終わるまでにしてほしいことを頼んできたし」 “The boss asked me to do something that he wanted me to finish by the end of the business hour”
*Btw, 「(頼んで)きた」(~してくる) is used when the subject does something to the speaker (that the speaker did not ask for).

If you’d like to make yourself subject:「部長に仕事が終わるまでにしてほしいことを頼まれたし」 “I was asked by the boss to do sth he wanted me to do by the end of the business hours.”

The rest part is good! ヤバいなぁ gives me an impression you don’t wanna miss sth live at home

This is perfect flawless Japanese! そうですね、この場合はbossの立場がわからないので、「上司」という言葉が一番しっくりくると思います

Nice! You can say 終業 but we tend to use it with 時間 or 時刻 more 「終業時間までに上司に仕事を頼まれたから」or you could say 「上司に終業時間までの仕事を頼まれたから」, I guess it sounds a little more natural

When we say “to be home late”, we often use「(家に)帰るのが遅くなる」as an idiom.
Your sentence already makes sense but it would sound more natural if you write 「上司に終業時間までの仕事を頼まれたから、家に帰るのが遅くなる。しょうがない…」

“My boss asked me to do something by the end of the day”

Cool that you chose the verb「欲しい」! I think 「…してほしい」grammar would be useful for your sentence 「(Aが)(Bに)…してほしい」: A wants B to do …

something: あること
by the end of the day: 今日中に

So it would be 「会長は私に今日中にあることをしてほしいです」

“I’m going to be home late.”

家に来る is “to come home” so 家に帰る “to go back to your home” sounds better here
「だから、家に遅く帰ります」 is better.
An additional expression you might want to learn is「家に帰るのが遅くなる」”to be home late”. As you are not on your way home yet, it sounds natural if you say 「家に帰るのが遅くなりそう」.
*「…そう」means “looks like something (someone) is going to do something.”

“Bummer…” You know 「…ったく」 expressionIt’s mainly used in verbal language (and rude) but I like the spirit! hahaha

Great! I always enjoy seeing this kind of freestyle (yet content-covering) translation

「もう～」sounds super proper here and I like it

For “something”, if the speaker knows what it is but avoids giving details, we use 「あること」(“a thing”)

For “the end of the day”, 今日中 (きょうじゅう) is a good expression It’s true that adding 「末」, as you used, is often used in Japanese business situations to indicate a deadline, but that’s mainly used after the month（月末）or year（年末）.

When we say “to be home late”, we often use「家に帰るのが遅くなる」expression. So you can say 「家に帰るのが遅くなっちゃう」 I really like that you used 「ちゃう」 for this translation! That suits this situation perfectly

終日 is an advanced word but it means “all day” or “throughout day” (I guess it’s close to “durante todo el día”) so 今日中 (きょうじゅう) sounds better

When we say “to be home late”, we often use「(家に)帰るのが遅くなる」as an idiom.
Other than that, looks great