0.5-th quantile or median, but other quantiles are also informative

Another word for quantiles is percentiles, so if you're used to thinking about test scores (my test score was in the xth percentile on that exam), then you already understand univariate quantiles.

On your first question, you're looking at what's happening toward the bottom of the distribution, the 10th percentile.

On your second question, I'm not sure if there's a right answer as Jiang Xuejun suggested, unless you have particular percentiles that interest you. Also, Roger Koenker, who co-created the method with Gilbert Bassett, also suggests looking at the entire distribution of percentiles, from 0 to 100 (see the example in Figure 4 of his JEP paper: http://www.econ.uiuc.edu/~roger/research/rq/QRJEP.pdf ).

Just as OLS is really a multivariate mean (ceteris paribus, how does variable x relate to variable y on average), quantile regressions provide a way to estimate multivariate percentiles/quantiles (ceteris paribus, how much does variable x relate to vary y at a particular percentile). OLS works best for the normal distribution (the BLUE property), but sometimes distributions are asymmetric, in which case quantile regressions inform you about specific points of the asymmetric distribution. Quantile regressions are also robust to heteroscedasticity, as there's a test you can perform based on equality of slopes of the quantile regressions; equal slopes across quantile regressions mean homoscedasticity.

Also, if you mean forecasting in a time series sense, I'm less familiar with quantiles in a time series context, though there are papers in that area, and panel regressions too. But whichever quantiles you select, you're talking about what's happening at the particular point in the distribution of your left hand side variable.

To make my answer more clear let us put our regression model, in general as:

y=f(x;B)+e

or

y=predicted values +residuals

When  predicted values less than y,   imply positive residuals, and the opposite is true.

As a result when the residuals are positive there is under prediction of the y values, and when the residuals are negative there is over prediction of the y values. Now, we go back to the quantile regression:

Quantile regression minimize a sum that gives asymmetric penalties (1-q)|e| for over prediction, and

q|e| for under prediction. As a result if q=0.1, for example, there is higher penalty for over prediction (0.9|e| ), and 0.1|e| for under prediction.

Based on the above clarification, your answer to the first question is correct, because negative residuals imply over prediction.

With regard to the second question, if you use 0.1 quantile it puts higher weight (0.9) to negative residuals, and lower weight to the positive residuals, but since the objective function in quantile regression is minimization, then the choice is the one with positive residuals.

If understood your question correctly, to avoid under prediction (because of higher cost) you should chose 0.9 quantile because your quantile regression minimization process is going to exclude the positive residuals in favor of the negative residuals.

OK, good luck with it

I think one should not mix-up regression (= statistical analyses) and forecasts. Quantils are one way to characterize a distribution. I do not know which data you have, but let us assume you have the electricity consumers ordered according to their electricity consumption for a series of years or quantiles derived from these data. Then you can e.g. find-out if trends of the quantiles are different and if and how variables like income or energy prices influence them. It is also possible that the quantiles you have refer to an other variable (for example to household income) and relates this quantile to electricity consumption. Such data one can analyse in a similar way as above (time series regressions).

On the base of statistical results, it should be possible to forecast the quantiles of a future distribution. Then one can decide (using usual test theory) which electricity consumption to forecast to be with a probability of, say, 90% sure not to be too high.

Best wishes

Anton