Will a 4K monitor display 1024x768 input signal?

probably, but I doubt it will look good.

The short answer is most likely, since that is a standard BIOS display resolution, but lessor signals are not guaranteed.  The built in image processor within the monitor needs to be able to accept that specific resolution.   Many of the generic models in particular have limited support for some resolutions.  The branded monitors typically have more robust image processors that can support more resolutions.   

some do, some don't. always read the specs first.

keep in mind, 4K monitors are 16x9, so even if it does display it, it will be stretched out and distorted.

sure.  any lesser signal will be displayed

You can do it but it wouldn't be a very pleasant experience.

Yes, but with a stretched or black barred display since is a different aspect ratio. 

Of course that's one of the lowest resolutions it will display it quite easily