According to your DXDIAG output, there is no free space on C:.
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Disk & DVD/CD-ROM Drives
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Drive: C:
Free Space: 0.0 GB
As the patcher uses the User temp folder on C: to unpack and install the data, before copying it to the final destination it requires, at least, 3GB of free space to run.
I also noted in the diagnostic section Windows update is also failing. That is probably where most of your free space went.
I am surprised AH3 would run at all. On the surface, with only 2GB of system RAM, and using the Intel video chip, which also steals memory from system RAM (up to 1GB), there would not appear to be enough resources to run the game.