I just want to know what words ancient Egyptians might have used for "country", "nation", "state", "kingdom", and "empire" (especially empire), preferably in their oldest forms. Can anybody help me out with that?
The Ancient Egyptians didn't really have words that quite matched those concepts. Their world-view was very different from ours.
Perhaps the nearest was the "hill country" hieroglyph (usually transliterated as 'xAst') which was often used as a determinative for foreign lands. This could also be spelled out, rather than using the single hieroglyph, for example as
These words might appear also in their own right in contexts where we translate them to mean "foreign land".
There are a few dictionaries of Middle Egyptian hieroglyphs available online, including this searchable pdf version which you might find of use.