* Have you tried holding Alt+SysRQ and typing REISUB when you're hung? If that works, your kernel's definitely still running so it's 100% a user space hang issue.
* Do you have any swap space at all on the system? If not, you're almost certainly running out of memory, which is likely causing unpredictable behaviour and process crashes, which would explain the system no longer responding. With 2GB you should probably have 4-6GB or so of swap; things will still be slow as you'll be swapping a lot but you shouldn't hang completely as before.
* Have you tried a lighter desktop environment? Ubuntu defaults to Unity which is a rather heavy DE in terms of resource usage. GNOME, MATE, Cinnamon, and KDE are similarly demanding; on such a low end system I'd suggest trying a lightweight DE like LXDE or XFCE, which will use a lot less system resources (don't worry, applications will still work as long as you have their dependencies installed - you don't have to be in a KDE session to run Qt apps, nor a Unity/GNOME session to run Gtk+ ones).
Thank you for your detailed reply!
To answer your questions:
1. Yes. Alt+SysRQ+REISUB seems to still shutdown the computer.
2. Yes. 2GB swap memory was installed by the default Ubuntu installation.
In addition, I have verified I have "en_US" locale installed that is used by the GOG.com installer script, and I have installed the package "gtk2-engines-pixbuf" that is also required by the installer. After these procedures, this is what I get when I run the TIS-100 installer from the command line (in the case the operating system manages to recover from the freeze without shutdown):
Verifying archive integrity... All good.
Uncompressing TIS-100 (GOG.com) 100%
Collecting info for this system...
Operating system: linux
CPU Arch: x86
trying mojosetup in bin/linux/x86
Error: Couldn't run mojosetup
Here is an example of how the "top" command looked after one event of a failed attempt to install TIS-100 (also, when the operating system managed to recover without shutdown):
top - 09:59:37 up 8 min, 1 user, load average: 0.18, 1.13, 0.77
Tasks: 192 total, 1 running, 191 sleeping, 0 stopped, 0 zombie
%Cpu(s): 2.5 us, 1.5 sy, 0.0 ni, 95.8 id, 0.2 wa, 0.0 hi, 0.0 si, 0.0 st
KiB Mem : 2052612 total, 1073820 free, 437292 used, 541500 buff/cache
KiB Swap: 2084860 total, 2084636 free, 224 used. 1145268 avail Mem
...doesn't seem to me like the memory was that big an issue, since not much data has been written to the swap memory at least?