Mechanical cable-actuated disc brakes started working really badly Currently I have a cyclocross bike with Tektro Lyra mechanical cable actuated disc brakes with 160mm wave rotors.

I have had issues with the brakes for some time now, mainly my fault, I accidently put some oil on the rear brakes so they almost stopped working (it was a while ago now). After properly cleaning them they seem to just slip so I ordered new brake pads. 

The current problem is not the pads or oil; it's the actual cable and braking mechanism (in my opinion): when I try to brake, the cable and lever are not tight enough, but when I tighten them it becomes really hard to pull the lever. Then I checked the brakes, and compared to the front brakes, they do not 'restore' to same place - when I brake, the lever is hard to press, but then afterwards the cable does not go back to its initial place for some time. The front brakes are super easy to pull, brake fast and restore to initial position almost instantly.

What could this be? I know it's hard to explain, but it seems like the cable is moving back really slowly, and the mechanism in the rear brakes that is supposed to snap back is not as effective.

Should I just take the bike to be repaired?
My guess would be the cable is old. The inner cable begins to seize or bind with the outer cable. This could be caused by corrosion caused by water ingress over time or dirt ingress over time. 

The binding basically causes more friction along the line of the cable - eventually preventing a smooth return to the "Off" position on the brake lever. 

You can do a "bodge" and that would be to squirt something like WD40 down the cable line to free it up. But this is only a temporary fix.
The proper fix is to install some quality pre-lubed cables which are also weather sealed.

Incidentally, pads contaminated with oil can be cleaned by removing the pads and heating the pads over a flame to burn off the oil.