Question 1 : My last example shows that the nodes are correctly centered and in your examples, if you use the option draw, you can see that the nodes are correctly centered too. The anchor used is (center) but if you put a text inside your node, the position of the text depends on the baseline.
Question 2 : It's easy to place a drawing exactly at the center of a node but a character is not symmetric and the baseline depends on the character. Your problem here is to determine if your character is symmetric and where is the baseline.
If you look at the last circle, $\circ$ is not correctly centered. In the four try the box with p$\circ$b is correctly centered vertically but not horizontally.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw [very thin, gray] (-1,-1) grid (5,1);
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (0,0) {$\circ$};
\draw [brown] (0,0) circle (3pt);
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (1,0) {p$\circ$ };
\draw [brown] (1,0) circle (3pt);
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (2,0) {$\circ$b };
\draw [brown] (2,0) circle (3pt);
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (3,0) {p$\circ$b };
\draw [brown] (3,0) circle (3pt);
\node [blue,draw,circle,minimum size=10pt,anchor=text] at (4,0) {p$\circ$b };
\draw [brown] (4,0) circle (3pt);
\end{tikzpicture}
\end{document}
I built a box (here a square correctly centered). It's more easy to see that the square is correctly centered. The last picture too but not $\circ$. When you used a character, this character is in a box with a baseline and the position of the character inside the node depends of the baseline.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw [very thin, gray] (-1,-1) grid (5,1);
\def\tvi{\vrule height 4pt depth 4pt width 8pt}
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (0,0) {\tvi};
\draw [red] (0,0) circle (3pt);
\def\tvi{\vrule height 4pt depth 4pt width 1pt}
\node [blue,draw,circle,minimum size=10pt,anchor=center] at (1,0) {\tvi$\circ$\tvi};
\draw [red] (1,0) circle (3pt);
\end{tikzpicture}
\end{document}

\fbox{A \textbullet A} \fbox{A $\circ$ A} \fbox{B} \hruledoes not seem to exhibit the problem illustrated by Depth of the\timessymbol. Surprisingly, using{$\times$}as the node text seems to get the center crossing exactly at the node's coordinate. – Peter Grill Jun 27 '12 at 02:39