No, the South Korean SIM card would have a South Korean phone number associated with it. To call/text her you would have to use that number. If you ALSO have an iPhone, iMessage would still work the same as her AppleID would still be the same. If you use texting instead of iMessage, it would be an international text, though, which is included in many plans.
Calling would mean international charges though, for both of you. You could always use Facetime or Skype over wifi to get around international calling charges, though.
First, I want to know if she has an iPhone 5 (including 5c and 5s) or newer iPhone (I'm not sure in what section I'm right now). If she does, then she probably will be able to do it with no problem. Just make sure that there are GSM carriers in South Korea (that may be one of the few countries in the world that doesn't have GSM carriers, but I'm not 100% sure).
See, the reason that I say this, is that I'm not sure if Verizon's CDMA bands on the phone will be unlocked to use on another international CDMA carrier. GSM carriers are A LOT easier for exchanging phone service and SIM cards on phones than CDMA.
While Verizon may have SHORT TERM solutions, for longer periods of times(i.e. 4 months as the OP stated) there are MUCH less expensive alternatives.
You may want to read my post(s) from the other day where I compare Verizon's International options with the option I eventually chose to use, Google's Project Fi.
For what it is worth, South Korea is one of the countries which Project Fi has international service as required by the OP.
Read my other message to you.
Good news: I just found out that South Korea is phasing out their CDMA carriers in favor of GSM. Most carriers are already using GSM, in fact, in some cases GSM only. So, your daughter should have no problem slipping in a South Korean SIM card and using her iPhone as a "South Korean" iPhone as long as her iPhone is: an unlocked iPhone; or a Verizon iPhone 5, 5c, 5s or newer. If she doesn't have an iPhone (again, I'm not sure what section I am), she can use any unlocked phone with GSM bands.
It looks like South Korea finally saw the light and is getting rid of [removal required by the Verizon Wireless Terms of Service] CDMA.