They seem to feel as if you're putting them in backwards then they just go straight into the slot. Are DS games suppose to go smoothly in the 3DS or is it harder since its a different design?
This is normal, as my 3DS does the same thing, and it reads the game cartridges just fine. I guess, since this is a new design, it's a little more difficult.
Not really. The 3DS games have a little tab like looking thingy that sticks out that's why they fit in the slot easily than the DS games.