Rule of thumb for confidence level

1. Yes confidence level always depends on choice of a researcher (at least in principle). There is no reason to use $90%$, $95%$ or $99%$ level either - it is more or less just convention. For example, in physics they use three-sigma or five-sigma significance which corresponds to confidence levels of $99,7%$, $99.99997%$ (and even higher ones can be reported).

   However, this being said once literature settles at a particular convention you will be fighting an uphill battle. For example, you would have hard time publishing in physics results that are not at least $3sigma$ or $5sigma$ significant without very good explanation. In economics, usually people settle for $95%$ level. Just because you can pick any significance/confidence level that does not mean others will accept it.

2. They actually do explain why there is $66%$ confidence interval. According to Stock and Watson:

Also plotted are $pm 1$ standard error bands, which yield an approximate 66 percent confidence interval for each of the impulse responses.

When it comes to VAR impulse responses (and forecasting in general) it is conventional to plot $pm 1$ standard error bands. This is also often done when people estimate DSGE models so it is more or less conventional practice. This is ultimately arbitrary choice on which people somehow settled.

Also note reporting 1 standard error deviation error bars does not mean the variable was significant at only $66%$ confidence level - for example $hat{beta}=30$ with $SE(hat{beta} )=1$ would have $t$-stat $=30$, would be significant at any conventional significance used in economics (in any reasonably sized sample). That does not prevent you from constructing 1 standard error bars (i.e. $30 pm 1$).