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My processor supports PAE and NX, but VMs don't see it. See attached /proc/cpuinfo. These are also enabled in BIOS (I've checked). Even though these are also enabled in the VM's settings, the VM don't see them. Windows 7 won't install because it can't see PAE. Windows 8 won't install because it can't see NX. What's the issue here?

Please pick one VM and provide Minimum information needed for assistance.

Let's focus on Windows 7 guest, then.
I have VirtualBox 4.13.14_OSE r95030, which I downloaded as a binary through my distro's package manager. As the issue is while trying to install the OS, I don't have Guest Additions installed in the guest yet.
The host is: Arch Linux x86. It is kept up to date with Arch's repos. 3 GB RAM.
The guest is (will be): Windows 7 Ultimate 32-bit, with 1GB RAM available.
Attached is the log from the last session trying to install the guest. I know the log says the extension pack is a different version than VirtualBox. I don't know what to do with that; my package manager says differently, and that everything's up to date. I know the log says the guest's HDD has a file system of ext4. I don't know what to do about that; I have done nothing with the VM except create it and the hard drive with the default settings (save for the RAM) and try to boot the Windows 7 install CD.
Also attached is a screenshot of the error.

The above simply will not work. The extpack must match the installed version of VBox.

This says that the host has PAE/NX but the guest has not been set to use it yet. Look in the processor setting for the guest and make sure that pae/nx is enabled.

If you take care of all of the above and still have issues you will need to ask the Arch folks what to do. We don't support their fork.

Well, that didn't fix it, so I guess I'll have to turn to Arch's support.