The 1st and 3rd equation cannot be solved since any rotation of a vector conserves the length of the vector, however, the length of {1,0,0},{1,1,0} ,and {1,1,1} are obviously all different.

For the 2nd equation the rotation matrix corresponds the unity matrix and different choice of angles produce a unity matrix. The natural choice would be all three angles equal zero (identy operation by doing no rotation)

For your first and third problem both Solve and Reduce are convinced there are no solutions.

For your second problem Solve cannot crack it, but Reduce is able to do it

TM={{Cos[phi]*Cos[si]-Cos[theta]*Sin[phi]*Sin[si], Cos[si]*Sin[phi]+Cos[theta]*Cos[phi]*Sin[si], Sin[theta]*Sin[si]},
{-Cos[theta]*Cos[si]*Sin[phi]-Cos[phi]*Sin[si], -Sin[phi]*Sin[si]+Cos[theta]*Cos[phi]*Cos[si], Cos[si]*Sin[theta]},
{Sin[theta]*Sin[phi], -Cos[phi]*Sin[theta], Cos[theta]}};
Simplify[Reduce[TM.{1, 0, 0}=={1, 0, 0} && 0<=phi<2Pi &&
-Pi/2<=theta<Pi/2 && 0<=si<2Pi,{phi, theta, si}],
  0<=phi<2Pi && -Pi/2<=theta<Pi/2 && 0<=si<2Pi]

which quickly returns

(phi == 0 && si == 0) ||
(phi == Pi && si == Pi) ||
(theta == 0 &&
 ((si + 2*ArcTan[Tan[phi/2]] == 2*Pi && 0<phi<Pi]) ||
  (si == -2*ArcTan[Tan[phi/2]] && phi > Pi)))

Checking each of those potential solutions, and many more, all return True

TM.{1, 0, 0}=={1, 0, 0}/.{phi->0, si->0}
TM.{1, 0, 0}=={1, 0, 0}/.{phi->Pi, si->Pi}
TM.{1, 0, 0}=={1, 0, 0}//.{theta->0,phi->Pi/2, si->2 Pi-2*ArcTan[Tan[phi/2]]}
TM.{1, 0, 0}=={1, 0, 0}//.{theta->0,phi->Pi/8, si->2 Pi-2*ArcTan[Tan[phi/2]]}
TM.{1, 0, 0}=={1, 0, 0}//.{theta->0,phi->3Pi/2, si->-2*ArcTan[Tan[phi/2]]}
TM.{1, 0, 0}=={1, 0, 0}//.{theta->0,phi->9Pi/8, si->-2*ArcTan[Tan[phi/2]]}

If your configuration implies there must be a unique solution for each of those problems then the only thing I can imagine is that some flaw has crept into the the construction of the problem.