Keep in mind that the p-value depends on the sample size. I guess you have a large sample size thus both values are p < 0.001. My interpretation would be that A is correlated with B and less so with C. Try to compare the partial correlation results with the normal correlation between A and B/A and C.
Note that correlation conditional on $Z$ is a variable that depends on $Z$, whereas partial correlation is a single number.
Furthermore, partial correlation is defined based on the residuals from linear regression. Thus, if the actual relationship is nonlinear, the partial correlation may obtain a different value than the conditional correlation, even if the correlation conditional on $Z$ is a constant independent of $Z$. On the other hand, it $X,Y,X$ are multivariate Gaussian, the partial correlation equals the conditional correlation.
For an example where constant conditional correlation $\neq$ partial correlation: $$Z\sim U(-1,1),~X=Z^2+e,~Y=Z^2-e,~e\sim N(0,1),e\perp Z.$$ No matter which value $Z$ takes, the conditional correlation will be -1. However, the linear regressions $X|Z$,$Y|Z$ will be constants 0, and thus the residuals will be the values $X$,$Y$ themselves. Thus, the partial correlation equals the correlation between $X$,$Y$; which does not equal -1, as clearly the variables are not perfectly correlated if $Z$ is not known.
Apparently, Baba and Sibuya (2005) show the equivalence of partial correlation and conditional correlation for some other distributions besides multivariate Gaussian, but I did not read this.
The answer to your question 2 seems to exist in the Wikipedia article, the second equation under Using recursive formula.
Best Answer
For understanding this I always prefer the cholesky-decomposition of the correlation-matrix.
Assume the correlation-matrix R of the three variable $X.Y.Z$ as $$ \text{ R =} \left[ \begin{array} {rrr} 1.00& -0.29& -0.45\\ -0.29& 1.00& 0.93\\ -0.45& 0.93& 1.00 \end{array} \right] $$ Then the cholesky-decomposition L is $$ \text{ L =} \left[ \begin{array} {rrr} X\\ Y \\ Z \end{array} \right] = \left[ \begin{array} {rrr} 1.00& 0.00& 0.00\\ -0.29& 0.96& 0.00\\ -0.45& 0.83& 0.32 \end{array} \right] $$ The matrix L gives somehow the coordinates of the three variables in an euclidean space if the variables are seen as vectors from the origin, where the x-axis is identified with the variable/vector X and so on.
Then the correlations of X and Y is $\newcommand{\corr}{\rm corr} \corr(X,Y)=x_1 y_1 + x_2 y_2 + x_3 y_3 $ and we see immediately it it $\corr(X,Y)=-0.29 $ because of the zeros and the unit-factor. We see also immediately the correlation $\corr(X,Z)=-0.45$ again because of the zeros and the unit-cofactor. However, the correlation between Y and Z is $\corr(Y,Z) = -0.29 \cdot -0.45 + 0.96 \cdot 0.83$ The partial correlation (after X is removed) is that part for which no variance in the X-variable is present, so $\corr(Y,Z)._X = 0.96 \cdot 0.83 $. Now imagine, the value $0.83$ would be $-0.83$ instead. Then the partial correlation would be negative and the correlation between Y and Z were $ 0.29 \cdot 0.45 - 0.96 \cdot 0.83$
What we see is, that the partial correlations are partly independent from the overall correlations (though within some bounds)