We just got a new door installed by a professional and there seems to be a bunch of leaks in the door. I am not sure if this is normal or not on a new door install and if they are easy to fix myself or they are not a big deal.
Your second image is a pretty good indicator that this is not normal. You can see rubber insulation at the end of the door. This insulation serves no purpose if it is not compressed against another surface. Light shining through is only a symptom of the problem. The fact that the right (image angle) side of your wall has an extrusion is of course a bit complicated, but simply cutting it out of the door (with a very generous offset) is definitely not the right way to approach this, especially since it is part of the frame which is static and does not need to move.
Thus answering your question if this is normal: Other countries might have different standards, but in my country I would absolutely not accept this kind of work. When I order a door I expect it to be perfectly sealed unless I tell them otherwise.
For your other question if you are able to fix this. It is hard to tell from the image, but the door seems to be at least level, so the offset between the door and its frame on the closing side seems to be constant. The simplest solution would be to remove the rubber insulation and replace it with a larger diameter rubber insulation. Since the pretty large gap on the (image angle) right part of the door does not need to move at all any kind of fixed insulation method (i.e. spraying some foam and cutting it and then paint over it) should probably do the trick.
edit: based on your 5th image, why is the gap on the right side a leak issue? From the outside it seems that the frame is properly sealed against the rest of the house. It looks horrible from the inside but leakage should not happen there.