You should be able to turn the camera in any direction - left, right, upside down- without experiencing this problem. The camera meters through the lens, so I'm not sure what you mean when you say that the lens is blocking the sensor.
The images you sent are all too different to effectively judge why you're experiencing this problem. I looked at the first three images and found that the exposure compensation (-2/3) and the ISO (800) are the same, but everything else is different. The shutter speed, aperture, focal length, time of day, angle to the sun, etc. are all different. For instance in one of the dark pictures the sun is coming from behind the little girl directly toward the camera. In another picture the sun is coming from behind the camera toward the child.
Try taking a few pictures in vertical and horizontal orientation of the exact same scene at the same time of day. Pick a simple background, like a wall, that's uniform in color and that will fill the frame in either orientation. Place something in front of the wall and photograph it dead center in both orientations. Take your time and make sure that you're not accidentally flipping any switches, pushing other buttons or changing any settings between shots. Focus on the item dead-center in the frame. Then post those images and we'll be able to help narrow down the problem.
By the way, I just noticed that in the EXIF date of the first two underexposed images the creation software is "Paint Shop Photo Album v5.21", but the third photo (the one that looks okay) the creation software just says "Ver.1.10. So, it appears that you used different software on these images. Could that have something to do with it? Do the vertical images look darker than horizontal images on the camera's screen (before you download them to the computer)?