So let's do these one a a time ....
Why is it that photos can compensate for light level and videos cannot?
They do, but not in the same way. When you take a still photo, the camera uses it's sensor to display the way an image will be saved. You see, a camera phone is basically a lens and a sensor. Unlike traditional film cameras, there is no shutter, or iris and the film speed it determined by the size and sensitivity of the sensor. So, when you take a picture, you are simply turning the sensor on so it captures the intensity and color of the light of each pixel. Then software takes over and balances it based on what the human eye finds most pleasing, converting the image to whatever format the software tells it to.
Videos do the same thing, but because a video is simply a lot of still images presented in sequence, in order to do that for EVERY frame, you'd either need symmetrical sensors (one for reading and one for capturing) or exposure is determined by key frames. Key frames are a frame in a video sequence chosen to determine the subsequent frame's exposure until the next key frame. If it's implemented well, it produces decent results. Of course if the lighting conditions are unusual, like low light situations or rapidly changing levels of light, a typical phone may not be able to interpret properly or keep up with the changing exposures.
Is there a way to hold a phone really steady to make pictures more detailed and less blurry?
Either a tripod or a
gimbal. The first is a tried and true old school method, but you have to be stationary and carry it with you. A gimbal, on the other hand, is a high tech solution that provided gyroscopic stability to those with shaky hands.
Why are screenshots saved separately from DCIM?
DCIM stands for
Digital
Camera
IMages and since screenshots are not taken through the lens, they get saved elsewhere. If you want screenshots in your DCIM folder, I'm sure there's an app out there that can do that.
What are the best tools for a not even amateur to do some photo editing/improvement, such as reducing glare and removing unwanted elements?
On the phone or PC? Most phones do come with some basic image editing, but it's usually just hokey filters and some exposure/color balance. All of that you can to online with Google Photos in a web browser, too. For more advanced users there's the grandaddy of them all ... Photshop. but that's expensive and complicated. You could try the free alternative called
Gimp which is a favorite of Linux users.
You could check out some of
these reviews, but to do any decent photo editing, you're going to need to understand at least the basics of image construction and manipulation.