きている vs. きました

In lesson 2.4 one person asks if they are here on vacation:

Also, just a side question, why is the particle "の" present in the first sentence? I know it's meant to show possession, but I don't see anything that can be possessed by anyone in that sentence. Or am I just reading it wrong and it's not being used as a particle in this sentence? 

I don't see why you couldn't use 「お休みで​来ましたか」 but I think Sayaka uses ​ 「来ている」 because she is asking about what Kenny is currently doing in the present—"coming for vacation?" Kenny responds using past tense, 「来ました」, because he has completed the movement, he "came" to Japan (for work).

They talk about the grammar point at about the 11 or 12 minute mark in the Rocket audio, but they don't go into great detail. English speakers do something similar so I didn't give it much thought. (Example: "Are you moving to California? No, I came to California for a friend's wedding.")

［~のです」 is an 'explanatory' phrase. Rocket doesn't cover it in Level 1 as far as I know but you can read more about it here:
http://maggiesensei.com/2010/09/08/request-lesson-when-and-how-to-use-んn-のです→んです%ef%bc%89/
 

The first one appears to be asking in present tense - お休みで​来ているのですか - coming and presently doing.

Where as this is past tense - お休みで​来ましたか

In the above scenario the first one is correct as it is asking what is presently happening.

The particle の is referring to the person being asked the question as in are "you" here on vacation. 

This one here お休みで​来ましたか I have a problem with. It does not strike me as correct. It seems too formal for speaking in Japanese and does not seem natural of what they would say.