Your system reduced to the essentials
p1 = {x2 - x1, y2 - y1, z2 - z1};
p2 = {x2 - x3, y2 - y3, z2 - z3};
p3 = {x4 - x3, y4 - y3, z4 - z3};
p4 = {x1 - x4, y1 - y4, z1 - z4};
list = {(p1 - p2) . {a2, b2, c2} == 0,
p1 . ({a2, b2, c2}*p2) == 0,
a2*x2 + b2*y2 + c2*z2 + d2 == 0,
(p2 - p3) . {a3, b3, c3} == 0,
p2 . ({a3, b3, c3}*p3) == 0,
a3*x3 + b3*y3 + c3*z3 + d3 == 0,
(p3 - p4) . {a4, b4, c4} == 0,
p3 . ({a4, b4, c4}*p4) == 0,
a4*x4 + b4*y4 + c4*z4 + d4 == 0,
(p1 - p4) . {a1, b1, c1} == 0,
p1 . ({a1, b1, c1}*p4) == 0,
a1*x1 + b1*y1 + c1*z1 + d1 == 0}
seems to be not solvable. GroeberBasis yields {1.} that is, there is a solution if 1==0
The equalities for the angles imply equations of the fourth order since
a^2 b^2 Cos[a,b]^2 == (a.b)^2 == c^2 d^2 Cos[c,d]^2 == (c.d)^2
