On the second level, is the stair closer to the bigger bedroom or the smaller bedroom? How can you tell from the drawing?
As you're going up the stairs, you're facing the 10x12 bedroom. The hatching on the left end of the stairs on the lower-level indicates that it's not really part of that level. It also wouldn't make sense for you to have to go through one bedroom to get to the rest of the upstairs.
Is the "deck" on the back of the apartment?
Yes. The front is at the other end. See the next item.
Is the front door between the kitchen and the closet?
Yes. It's shown in the open position in the floorplan. I've circled it here:
Does "duplex" here mean that it has two levels?
A duplex usually means that there are two separate dwelling units in a building; there's probably another identical or mirror-image apartment on the other side of one of the long walls.