Window film affixed to the wall around a window?
You'll be just fine sticking the tape to the wall outside the frame then sticking the plastic to that. There's a reasonable chance that there is cold air moving through gaps in the exterior trim, along the outside of the window casing, then through gaps in the interior trim. Applying plastic as you've suggested will help reduce this cold air movement.
WARNING science content: It's actually the warm air that's moving out, not the cold air moving in. There's more energy in warm air, so it's moving faster and wants to escape to where the temperature is colder in order to equalize temperatures. Also, your HVAC system is forcing warm air into the house, slightly increasing the interior air pressure. That increase in pressure wants to escape to where the pressure is lower to balance things out, no matter what the temperature is. Except, of course, when the wind is blowing outside and forcing the cold air in because now the outside pressure is higher.
Unless there is zero insulation in the wall (and even if that is the case), it's highly likely you're losing more heat through the glass than through the wall.
Warning: Do be aware that the double-stick tape used to hold the window insulation film in place can remove paint or wallpaper from whatever it's attached to (window frame or wall). Be prepared to repair damage or sacrifice some of your damage deposit if this happens.

Just a thought for you: With all the issues you seem to be having with insulating your windows (and exterior walls), if your landlord doesn't show any interest in improving the situation, you may want to consider moving as soon as your lease is up! I realize that economic realities may make that difficult, so don't feel you need to address this point in a comment, it's just something for you to think about.