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I exported a pdf from ArcMap. I opened it in Adobe reader. I can measure and calculate perimtere, but calculate area result is many times 0. Why?

Also, I have TerraGo installed in Adobe Reader, trial version, but none of the options from GeoPDF or GeoMark are enabled and at GeoPDF File Properties it says it is not an actual GeoPDF. Why?

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A GeoPDF is a specific file format falling under the broader category of geospatial pdfs. They are not the same thing in much the same way that a GeoTIFF differs from a regular TIFF - internal structure and formatting differ.

If you used File > Export to create your pdf, you might have created a geospatial pdf as Esri does have some support for this. However it is limited and mostly tied to the Adobe Acrobat 9 specifications (per the second link, Adobe have their own ideas proposed to an ISO standard). The file created is not a GeoPDF; for that, you would use to need a pdf writer/exporter compliant to or created by TerraGo's standard, such as their Publisher software. And since you haven't created an actual GeoPDF out of Esri with their extensions, their Acrobat Reader extensions don't have anything to read (the two software extensions are something of a matched pair).

This is also discussed in a few other questions/answers and their comments here, for instance TerraGo geopdf vs. Esri geopdf

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