Is sampling at double the desired reproduction frequency accurate?
There is not only one sampling rate that theoretically allows exact reproduction. The sampling theorem states that it is sufficient for perfect reconstruction if the sampling frequency $f_s$ is greater than twice the highest signal frequency:

$$f_s&gt;f_{max}\tag{1}$$

Without further restrictions, Eq. $(1)$ is also necessary for baseband signals occupying the whole frequency range $[0,f_{max}]$. Some people like to discuss whether in $(1)$ there should be a "greater or equal" sign or just a "greater" sign. In practice, this question is totally irrelevant because you always have to sample at a frequency twice the maximum frequency plus some extra tolerance to allow for non-ideal filters.

As you've noticed, sampling a sinusoid at exactly twice its frequency will generally not allow you to reconstruct it. As soon as you sample at a rate slightly larger than twice its frequency you will be able to reconstruct it perfectly.

Note that there is band pass sampling for band pass signals where the sampling rate can be considerably smaller than twice the highest signal frequency for perfect reconstruction to be possible. However, it's generally not true that sampling at twice the signal's bandwidth is sufficient, even thought that may be the case for specific signals.

These are only the basics of (uniform) sampling of 1-dimensional signals. There is much more to be learned. Browse the "sampling" tag on this site, and read the overview article Sampling - 50 Years After Shannon by M. Unser.