Let's analyze each option from List-I with their corresponding nature from List-II:

1. Series A: \(12 - 7 - 3 - 2 + 12 - 7 - 3 - 2 + \dots\)
   This series is repetitive, creating a block pattern of constant terms that repeat indefinitely. It is not convergent or divergent because it doesn't show a consistent trend towards infinity or a finite value. This makes it oscillatory.
   Match: II (oscillatory)
2. Series B: \(1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots\)
   This is the Alternating Harmonic Series, which is known to converge to a finite value due to its alternating signs and decreasing terms. It is conditionally convergent, as it only converges due to the alternation.
   Match: IV (conditionally convergent)
3. Series C: \(\sum_{n=0}^{\infty}\{(n^3+1)^{1/3}-n\}\)
   Approximating the terms as \((n^3+n)^{1/3} - n\) for large \(n\), this simplifies to a positive term \(\frac{1}{3n^2}\), which diverges because the summed terms do not tend to zero quickly enough.
   Match: III (divergent)
4. Series D: \(\sum_{n=1}^{\infty} \frac{1}{n(1+\frac{1}{n})}\)
   This series simplifies to \(\frac{n}{n+1}\), which tends to 1 as \(n\) approaches infinity. Written as \(\frac{1}{1 + \frac{1}{n}}\), this is similar to the harmonic series, but each term is slightly reduced causing convergence due to the added structure.
   Match: I (convergent)

Therefore, the correct option is:

A-II, B-IV, C-III, D-I