There is no distinction between an equiangular/ equilateral/ regular triangle.
True ; False
True. For the triangle (and only for the triangle) these properties all hold for the same figure.
2.
There is no distinction between an equiangular/ equilateral/ regular quadrilateral.
True ; False
False. A rectangle is equiangular, a rhombus is equilateral, and a square (which is a special type of rectangle and rhombus) is the only regular quadrilateral.
3.
Given (only) that the tangent of an integer lattice angle is a rational number, you could show that an equilateral triangle with vertices in the integer lattice exists.
True ; False
False. The tangent of the triangle angle /3 is 3 which is not a rational number.
4.
Any regular lattice polygon with vertices in the integer lattice must have sides of integral length.
True ; False
False. For example the lattice square with vertices O, (1, 1), (2, 0), (1, –1) has sides of length 2. However, it is true that the square of the side length is an integer.
5.
An equilateral triangle can be embedded in the equilateral triangular lattice in only one way.
True ; False
False. Just as the square can be placed in the integer lattice in many ways, so there are many ways in which the equilateral triangle can be placed in the equilateral triangle lattice.
6.
Explore to see if Scherrer’s proof can be extended to cover equilateral or equiangular lattice polygons.
No, it fails in both cases. The smaller pentagon in the figure is no longer similar to the original, and does not have the same properties.
/3 is 
3 which is not a rational number.