English is unique in that "come" and "go" do not relate as much to direction of travel as your perspective when you speak. Perspective is where you imagine you are when you are speaking. For example, suppose I'm talking to my friend on the phone:
I am leaving right now to come to your party.
I'm not at my friend's party, but I visualize that I am at the party with my friend. In a similar way:
Would you like to come to my friend's party with me?
Neither of us are at the party, but by using "come" instead of "go" it's an invitation that visualizes us both at the party. Of course I could also ask:
Would you like to go to my friend's party with me?
This is perfectly grammatical. The only difference is my perspective of us, here, moving toward the party together.
So the answer is all of your sentences are correct, and most of the time it doesn't really matter where I am when I say them. "I come to work", "I go to work" -- either way I don't have to be at home or at work to say these. It all depends on my perspective, whether I imagine myself already at home or already at work.
Keep in mind you have to use the perspective that makes sense in context. If my friend and I are thinking of a trip to Europe, I would say
Let's go to Europe.
and not
Let's come to Europe
However if I was talking to my friend in Europe I might say,
I want to come there to see you!
The difference is that, in the first case, there is no one in Europe whose perspective I relate to. In the second case I can put myself in my friend's shoes and see myself coming toward her. However, I can ask my friend in the US:
Do you want to come with me to Europe?
because I visualize that person with me as we move toward a location -- she's "coming along" with me.
It may sound confusing at first, but it does make sense once you get used to it.
No, they do not mean the same thing. Not only does around make the second sentence grammatical, but the use of for in the first sentence gives that sentence a different meaning.
He has failed the exam for the first time.
He may have written the exam five times before. But this time, the sixth time, he has failed it. It is the first time that he has failed it.
He has failed the exam the first time around.
This means that it's the first time he's written the exam—and that he's failed it.
The sentence construction is a bit odd, but it works if you think of somebody announcing the result immediately after the attempt—and just before he attempts it a second time.
Note the difference that around makes in variations of the sentence:
✔ He has failed the exam the first time around.
✘ He has failed the exam the first time.
✔ He failed the exam the first time around.
✔ He failed the exam the first time.
In the has failed version, the sentence becomes ungrammatical if you simply remove around. However, I can't really explain why this is the case.
The final version without around seems correct, but it is also awkward; I would add he wrote it to the end.
Best Answer
You cannot use the present perfect with a time phrase that excludes the present, which the first time does, unlike for the first time.
"The first time" requires the past tense.
P.S. the first time refers to the first of several or of many, whereas for the first time refers to the first as first, as something new.