The key to this problem is that you say the chain gets slack when you backpedal. This indicates with some certainty that your freewheel is the problem. To dispense with all doubt, remove the chain entirely from the gears on your rear wheel and spin the gears with your hands. It should spin easily and make a satisfying clicking noise. Any resistance there is likely the culprit. If you're unable to test it this way, try just pushing your bike along with your hands and notice if the pedals turn with the wheels on their own, as if a ghost were riding your bike; that is an indication of your freewheel being the problem.
If the freewheel doesn't spin easily or you notice a ghost riding your bike when you push it, you will most likely have to replace the freewheel. These parts do respond very well to oiling, but the gap to oil them is on the backside, closest to the spokes, requiring that it be removed, and removing it requires special tools. I imagine that after a year of regular use, this part is almost certainly worn out in the normal sense and ought to be replaced anyways. These parts aren't too expensive on their own, just keep in mind that you will also have to replace your chain. This is NOT just a way to up-sell you parts! These two parts need to mate perfectly with each other to work; replacing one without the other will leave gaps and it will not work. These are considered wear-and-tear and will need replacing often.
However, if the freewheel spins easily and doesn't seem like the issue, the only other place I could foresee a problem is in the jockey wheels of your rear derailer. The rear derailer is the elbow-shaped device that moves the chain back-and-forth to different gears. The jockey wheels are the two small gears that guide the chain. If these have gotten dirty or have broken teeth, they can prevent the chain from easily freewheeling.
Other things to consider: check your chainring (big gear by the pedals) and make sure it doesn't wobble and doesn't have any broken teeth. Also try to notice whether this problem happens on particular gears more than others. If so, it may be an unavoidable issue related to the bend that your chain needs to make to reach the gears you're using. You'll have to learn to shift into a middle-gear before backpedalling.
One final note: dropping chains is going to be a perpetual issue for your bike due to its design (single chainring, no catcher/front derailer). However, it should be a rare occurrence, and it should NEVER come off simply from coasting. I only mention it because it will be an occasional fact of life.