Yes, one of the reasons for using 2-layer PCBs is to avoid the jump wires. On a complex board there would be lots of them in all directions. Your layout doesn't need the second layer, though; it's still simple enough to be routed on a single layer without wire bridges. If you get stuck you may have to move some components to another position, but it can be done.
I would suggest to try it instead of going the easy way of the 2-layer board. It's about practice. If you don't succeed in doing this on a single layer, you'll never be able to do a more complex board on a 2-layer PCB either.
A professionally made multilayer PCB will have copper-plated vias to connect the top trace with the bottom trace. If you plan to make the PCB yourself you won't have the through-metallization, but since you only use PTH parts you can solve that by soldering the wires on both sides of the board. For some components the wire may not be reachable on the component side. Just make sure there's no trace to the hole on that side then. If you would need vias apart from the components' mounting holes you can solder a thin wire on both sides of the hole.
2-layer PCBs will often use both layers for routing, and even component placement. On densely populated boards this may not leave much room for copper planes, like for ground or Vcc. If these are really required designers often will go to a 4-layer board. A ground plane not only helps in the design of HF designs, but ground is often the net with the most connections, so it takes a lot of routing away from the outer layers as well.
Today most PCBs use SMT technology, and SMDs will often be placed on both sides of the board. First the components on one side are fixed with glue dots, then the board is flipped and the parts on the other side are placed. The glue is needed to avoid the components falling of when the board is turned upside down. Some glue dot machines can place up to 50000 glue dots per hour, that's 14 per second!
Best Answer
Since you did not mention how many Layers your PCB is going to have, we have two scenarios.
If you got only a 1 layer PCB, then the best solution is to put the SMD parts on one side and the Pins on the other, so you can easily solder them to the pads. Side view on PCB:
This is obviously the best option as long as the parts on the other side are almost of the same height. Depending on how the PCB is going to be used, you might want to add legs.
Since your picture just shows the layers top (red) and pads (green), but no bottom layer (blue) make sure its checked and therefor visible (you might want to turn of the top layer, when routing the bottom layer).
Afterwards you should check whether you got the right library for the Pin Headers. There are definetly ones with pads on both sides of the PCB, for instace in the standard library "pinhead".
If you place a via in eagle (2 layer PCB) you always have connections on both sides. Make sure to check the "via" option if the vendor has one, because they are not always added by default.
EDIT:
I tested it in eagle and there is a connection between top and bottom layer of the PCB over the pinhead:
So if you order correctly, then there should be a connection between both layers over the pads on both sides. Just use the rule checks from eagle to make sure everything is connected (run ratsnest and turn off all layers except "Unrouted").