Inside or on the faces of a tetrahedron with five edges of length $2$ and one edge of lenght $1$, there is a point $P$ having distances $a, b, c, d$ to the four faces of the tetrahedron. Determine the locus of all points $P$ such that $a+b+c+d$ is minimal and the locus of all points $P$ such that $a+b+c+d$ is maximal.
1. **Volume and Face Areas Calculation**:
   Let \( V \) be the volume of the tetrahedron. We need to express the volume in terms of the distances from point \( P \) to the faces of the tetrahedron. Let \( A \) and \( B \) be the areas of the small and large faces of the tetrahedron, respectively.

2. **Volume Identity**:
   The volume \( V \) of the tetrahedron can be expressed using the distances \( a, b, c, d \) from point \( P \) to the faces. The identity \( 3V = A(a+b) + B(c+d) \) holds, where \( A \) and \( B \) are the areas of the small and large faces of the tetrahedron.

3. **Expressing \( a+b+c+d \)**:
   We can express \( a+b+c+d \) as a linear function of either \( (a+b) \) or \( (c+d) \). This is because the volume \( V \) is fixed, and the sum of the distances from any internal point to the faces of a tetrahedron is related to the volume and the areas of the faces.

4. **Extremal Values**:
   The sum \( a+b+c+d \) reaches its extremal values when either \( (a+b) \) or \( (c+d) \) is maximal or minimal. This is due to the linear relationship established in the previous step.

5. **Minimal Sum**:
   The sum \( a+b+c+d \) is minimal when \( P \) lies on the edge with side length 1. This is because the distances to the faces are minimized when \( P \) is closest to the smallest face.

6. **Maximal Sum**:
   The sum \( a+b+c+d \) is maximal when \( P \) lies on the edge opposite to the edge with side length 1. This is because the distances to the faces are maximized when \( P \) is farthest from the smallest face.

Conclusion:
The sum of distances \( a+b+c+d \) is minimal if \( P \) lies on the edge with side length 1 and maximal if \( P \) lies on the opposite edge.

The final answer is \( \boxed{ P } \) lies on the edge with side length 1 for minimal sum and on the opposite edge for maximal sum.