The short answer is 'NO'
All games you digitally purchase from the Nintendo eShop are tied to the Nintendo ID used for purchase. As soon as you change the Nintendo ID in your 3DS/3DS-XL, the old games become inaccessible. The only way you can regain access to them through your Nintendo ID would be to purchase them from eShop using your Nintendo account.
You will use the 3DS System Transfer from the System Settings -> Other Settings menu (sub selection/page 4).
The menu options should be the same on the New 3DS even if they're shuffled around.
The process will transfer everything over from the old 3DS to the new New 3DS except for anything that was a part of a pre-installed bundle (eg, if you bought a 3DS with Mario World pre-installed, that will not transfer over).
When the process is done, the old 3DS will be reset to it's original state as when you originally bought it. Note, that most DSi Ware save game data and some DLC (no list on hand but it can be re-downloaded) may not transfer in this process.
You will loose DSi Ware save game data and will have to replay those games to unlock stages etc. (some examples of this are Mario vs. Donkey Kong Minis March Again and Aura-Aura Climber).
I just recently did some 3DS system transfers of DSi -> 3DS and 3DS -> 3DS recently and talked to a Nintendo support rep to confirm some steps in the process and also talked about future New 3DS migration and I was confirmed that once the hardware is out in the region they will update the relevant support pages with any changes, so check there first before you do anything and when in doubt, call their support line.
NOTE: Relevant info from the Nintendo Support site. 1) do not setup your NNID on the new system before the transfer process. 2) If the SD card you have on your old system is a standard SD card that is greater than 4GB, you will need to buy a Micro SD card (with an SD card adapter) and using a PC, copy the contents of that SD card to the Micro SD card.
Best Answer
A 3DS system will take a micro SD card and automatically format it. There are no default files on the SD card, it's only used to provide extra space for storing save files. You can pull an SD card out of a 3DS and put it into a computer to look at it, there's nothing special there. You can also swap out SD cards like you would change a memory card for only PlayStation or Nintendo consoles, using multiple SD cards to store games with a single 3DS console. I've done this myself before when I had too many save files and ran out of space.
You can test that the SD card works by using settings to move files from the internal hard drive to the SD card, or saving games to the SD card directly. If those files still work, you'll know the SD card is all set to go.