1-- Cut it into thirds, fourths or quarters, fifths, sixths. 2-- As above.
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Thirds, quarters, sixths etc are plural nouns of the particular fraction, so we cut it into halves, thirds, fifths, sixths etc. "pieces" is plural, so we cut it into pieces (into two pieces, ten pieces etc). "Cut it
Technically you can;
"Bisect" it - cut it into halves
"Trisect" it - cut it into thirds
"Quarter it" - cut it into quarters
I don't think the terms pentasect, hectasect, septasect, octosect, nanosect or decasect actually exist (they are not in the Windows Spelling Dictionary anyway), so after four pieces go the long route; "Cut it into fifths" etc.