What is the model number on the sticker on the card?  Does it match what is on the box?  Is this what it say, "03G-P4-2784-KR"?  If so, it shouldn't affect your warranty as the card would match its serial number, but somehow was put in a wrong box possibly.
 
Also, changing thermal paste should not affect the warranty.  EVGA is typically one of the more lenient brands compared to the others such as ASUS or MSI. 
 
I would suggest registering your card to your account.  This way it would make any potential RMA issues down the road much easier.
http://www.evga.com/support/register.asp
 
Otherwise, enjoy your new card!

No problem, happy to help.  So is the model number end in -2782-KR or -3783-KR then?  The sticker on the back of the card will look like this --->
This part number should be on one of the side flaps on the box as well.  Do the serial numbers match between the box and card?
 
Regarding overclocking, perfectly fine to do that and won't void the warranty.  You can even overvolt within Precision X, although it is only between +38mV to +63mV which is a fairly low amount.

I've noted on your account about the issue, along with a link to this thread. It shouldn't affect your warranty at all since the card you bought was a reference 780 and the serial number is also for a reference 780. 
 
And just to make it official, replacing the thermal paste is fine, changing out the stock cooler for an afteremarket cooler or water block is fine, and overclocking is fine. Just don't physically damage or alter the card or flash a custom firmware to the card. If you do change out the cooler, keep the stock one in case you ever need to RMA the card, otherwise there will be a fee for replacing the cooler.