OK. When it overflows, it should overflow through the white plastic pipe on the right. If that joint is leaking it probably wants tightening up. But that pipe should be big enough to take all the overflow, the level should never rise high enough to reach the mounting screw holes in the back. Take some more photos of that overflow pipe please, close up, inside and out. perhaps it is blocked.
More modern flushing valves incorporate an overflow so that the water goes down into the pan. Your plumbers have used a more traditional type that relies on the overflow on the side of the cistern.
As to the overfilling, you see the upside-down blue cup. That's the float. The cup fills with air during flushing, so it rises with the water level and should press by a lever on the valve to cut off the water supply. The type you have is a copy of a Torbeck or Armitage Shanks equilibreum valve. A very good valve is some ways because it is very quiet. However it is prone to grit, dirt or scale jamming the small parts inside, which is probably what has happened in your case. I don't see scale so you are probably in a soft water area. Often dirt or rust in the pipes is dislodged when plumbers are working, so they are supposed to let water gush from the pipe before fitting the valve to wash it away. It is actually very easy to dismantle and wash out any grit, but unless you are fond of DIY plumbing I won't advise you to do that.
You can try gently tapping or pressing down the white lever where the stem of the float attaches, this will open and shut the valve and may make it temporarily seal better. You can also (gently!) screw the blue stem down so that the float goes lower. This will reduce the filling height and will make the float press harder on the valve if it overfills. An inch or so will do. It is essential that the float goes up and down freely and does not rub on the cistern or the flushing valve, which prevents it moving correctly.
As the parts in your cistern look quite new, I think it is a fitting error or grit, but if you show some more pics I can compose a short note that you can copy and send to the landlord or housing company. It will probably advise that the plumber should come back and rectify the work recently done.