Corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides trisected?

Yes. If we cut off corners to create a regular hexagon, then each angle of the hexagon is $120^\circ$, meaning that each angle of each removed triangle is $60^\circ$, so these triangles are equilateral.

Now all sides of the hexagon are equal. Each triangle you removed shares a side with the hexagon, so all its sides are equal to the side length of the hexagon. Thus the three parts of each side of the original triangle are equal - two of them are sides of removed triangles and the third is a side of the hexagon.

When you trisect the sides, you remove three equilateral triangles and the sides of the hexagon are equal.

In order to handle this question, be it about the size of the segments or the surface, don't look to this question as "cutting the edges of the equilateral triangle", but as "folding the edges of the equilateral triangle towards the centre": it produces exactly the same result (the equidistant hexagon), but the questions about segment length and surface become trivial.