A better expectation operator which plays nicely with undersetting?

You can use the model of \int for this:

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand{\E}{\operatorname*{\mathbf{E}}\ilimits@}
\makeatother

\begin{document}

\[
\iint_{R} f(x, y)\, dx \, dy \qquad \text{versus} \qquad \iint\limits_{R} f(x, y) \, dx \, dy
\]
\[
\E_{R} f(X, Y) \qquad \text{versus} \qquad \E\limits_{R} f(X, Y)
\]

\end{document}

Using \ilimits@ will make the command obey the intlimits option just like \int.

Use \nolimits instead of \ilimits@ if you don't want that.

For some reason, \DeclareMathOperator will not react to \limits. However, \DeclareMathOperator* will react to \nolimits. So this should give you the result you are lookiing for:

\DeclareMathOperator*{\E}{{}\bf E}

\[
\E\nolimits_{R} f(X, Y) \qquad \mbox{versus} \qquad \E_{R} f(X, Y)
\]

For technical reasons, I'm not able to show output. (Since retiring from AMS, I have lost access to all my previous resources, and haven't yet rebuilt a working system.)

The presence of the {} in the definition is intended to counteract the default positioning of a single character as an operator to be vertically centered on the math axis. This is the default (specified in Appendix G of the TeXbook) so that operators such as \sum, \int, etc., are properly positioned.

The inability of the unstarred \DeclareMathOperator to react to \limits seems unreasonable, so I will make a request for it to be implemented in a future release of amsopn.

Update. I have been urged to check the documentation in amsopn.dtx, and there I find that \limits has been purposely inhibited, as pointed out by @egreg. I therefore withdraw the suggestion that it be made available.

Your first operator works the other way:

\documentclass[preview]{standalone}

\usepackage{amsmath}
\DeclareMathOperator*{\E}{\mathbf E}
\DeclareMathOperator{\EE}{\mathbf E}
\newcommand{\EEE}{\mathop{{\mathbf E}}}

\begin{document}
\[
\iint_{R} f(x, y)\, dx dy \qquad \mbox{versus} \qquad \iint\limits_{R} f(x, y) \, dx dy
\]
\[
\E_{R} f(X, Y) \qquad \mbox{versus} \qquad \E\nolimits_{R} f(X, Y)
\]

\end{document}