I have thought of a situation where an emulator would help, and I am wondering what emulators exist that would fit these needs:

* Must be able to run the GOG version of Akalabeth
* Must have save state support (this is where DOSBox fails)
* Ideally, it should allow for the recording and playback of input sequences

So, any emulators that would fill these needs?

If all else fails, there's always VMware Workstation, or its free alternative VirtualBox.

As for recording and playing back input sequences I would suggest a macro, maybe TinyTask

I'm pretty sure you're asking something not possible in most cases, but maybe you could look into the tools TAS uses.