Here is what happens. I turn on the computer and it gives some sort of warning in white letters
in a red box that says that there is no partition. I figure that this is OK since that is how
I remember setting up the system. I thought that there was no issue with it having no partition
That makes absolutely no sense. There has to be a partition that you installed into. When you get an error, we need exact text. It is hard to help you when you only tell us the general theme of the error. You couldn't have installed to no partition, as the files had to go somewhere inside a partition. They can't exist in unpartitioned space.
So, thinking that the problem must be with the fact that my laptop has an ancient screen resolution,
I decided to connect a more modern monitor to the laptop and try again. This time, it pretty much the same
as it did before except for the very end there was a different error instead of a blank screen.
That won't work at all. You can't go from the laptop built in screen to an external monitor, because you need to configure X for a dual headed display. Just plugging in a monitor will only show you the boot, which you took images of. Once the boot is done and X starts, it will start on the laptop only. Fortunately, the last screenshot you posted showed the problem there. Your refresh rates are nowhere near where they should be.
Your horizontal frequency is set to 15.7 when it needs to be between 30-70, and vertical is set to 30 when it needs to be between 50-120. To change those values, edit the file /etc/X11/xorg.conf. Specifically, change the H and V refresh rates to those specified by the monitor, unless those were the external monitor, and not your laptop's. Get the correct values from the laptop manufacturer's website, or just go for a middle-of-the field value.
The next screen shows the menu for starting in single user mode:
That is called the grub menu, and shows exactly what we would expect. Grub normally gives the choice of default boot and single user. Single user is only intended to be used for fixing problems, not actual use.
Here are the following screens after I select single user mode:
All of those are the normal boot process.
Try setting your xorg.conf to better values, and see what happens. The blank screen you describe on a normal boot is actually exactly what should happen if you have bad resolutions. To the computer's point of view, everything is fine, which is why you get no errors. The problem is a conflict between the way things are set and the way they should be set.
You may find it best to boot into text (as mentioned earlier, hit ctrl-alt-f1, or f2), become root, and issue the command
That should give you the options of automatically detecting the default values for things like the refresh settings.
The real problem may be the partition issues.