Find all polynomials $P(x)$ of the smallest possible degree with the following properties:

(i) The leading coefficient is $200$;
(ii) The coefficient at the smallest non-vanishing power is $2$;
(iii) The sum of all the coefficients is $4$;
(iv) $P(-1) = 0, P(2) = 6, P(3) = 8$.
To find the polynomial \( P(x) \) of the smallest possible degree that satisfies the given conditions, we will follow these steps:

1. **Determine the degree of the polynomial:**
   - Since \( P(-1) = 0 \), \( x = -1 \) is a root of \( P(x) \).
   - The polynomial must also satisfy \( P(2) = 6 \) and \( P(3) = 8 \).
   - To satisfy these conditions, \( P(x) \) must be at least a polynomial of degree 4 (since we need at least 4 roots to accommodate the conditions).

2. **Formulate the polynomial:**
   - Let \( P(x) = 2(x+1) + (x+1)(x-1)(x-2)(x-3)Q(x) \), where \( Q(x) \) is a polynomial to be determined.
   - The term \( 2(x+1) \) ensures that the coefficient of the smallest non-vanishing power is 2.

3. **Incorporate the leading coefficient:**
   - The leading coefficient of \( P(x) \) must be 200.
   - The leading term of \( (x+1)(x-1)(x-2)(x-3) \) is \( x^4 \), so we need \( Q(x) \) such that the leading coefficient of \( 200(x+1)(x-1)(x-2)(x-3)Q(x) \) is 200.
   - This implies \( Q(x) \) must be a constant, say \( Q(x) = 1 \).

4. **Verify the polynomial:**
   - Substitute \( Q(x) = 1 \) into the polynomial:
     \[
     P(x) = 2(x+1) + 200(x+1)(x-1)(x-2)(x-3)
     \]
   - Expand and simplify \( P(x) \):
     \[
     P(x) = 2(x+1) + 200(x^4 - 6x^3 + 11x^2 - 6x)
     \]
     \[
     P(x) = 2x + 2 + 200x^4 - 1200x^3 + 2200x^2 - 1200x
     \]
     \[
     P(x) = 200x^4 - 1200x^3 + 2200x^2 - 1198x + 2
     \]

5. **Check the conditions:**
   - The leading coefficient is 200.
   - The coefficient of the smallest non-vanishing power (x) is -1198, which does not satisfy the condition that it should be 2.
   - Therefore, we need to adjust \( Q(x) \).

6. **Adjust \( Q(x) \):**
   - Let \( Q(x) = x + c \).
   - Substitute \( Q(x) = x + c \) into the polynomial:
     \[
     P(x) = 2(x+1) + 200(x+1)(x-1)(x-2)(x-3)(x+c)
     \]
   - Expand and simplify \( P(x) \):
     \[
     P(x) = 2(x+1) + 200(x^5 + (c-6)x^4 + (11c-6)x^3 + (11c-6)x^2 + (c-6)x)
     \]
   - To satisfy the conditions, we need to find \( c \) such that the coefficient of \( x \) is 2.

7. **Solve for \( c \):**
   - The coefficient of \( x \) in \( P(x) \) is:
     \[
     200(c-6) + 2 = 2
     \]
     \[
     200(c-6) = 0
     \]
     \[
     c = 6
     \]
   - Substitute \( c = 6 \) into \( Q(x) \):
     \[
     Q(x) = x + 6
     \]
   - The polynomial becomes:
     \[
     P(x) = 2(x+1) + 200(x+1)(x-1)(x-2)(x-3)(x+6)
     \]

8. **Verify the final polynomial:**
   - Expand and simplify \( P(x) \):
     \[
     P(x) = 2(x+1) + 200(x^5 + 5x^4 - 10x^3 + 10x^2 - 5x - 6)
     \]
     \[
     P(x) = 2x + 2 + 200x^5 + 1000x^4 - 2000x^3 + 2000x^2 - 1000x - 1200
     \]
     \[
     P(x) = 200x^5 + 1000x^4 - 2000x^3 + 2000x^2 - 998x - 1198
     \]

The final answer is \( \boxed{P(x) = 200x^5 + 1000x^4 - 2000x^3 + 2000x^2 - 998x - 1198} \).