Quality of water for drinking purposes
Does this mean that the higher the concentration of bicarbonates, the better the quality for drinking?


Not necessarily. This is more a subjective question involving what tastes good to you.


  Should we choose the bottle of water with the highest concentration of these anions?


Assuming the water is actually monitored for safety/quality, the upper limit on caustic (hydroxide) alkalinity is likely too low to do any damage. And carbonate alkalinity will be limited by pH and equilibrium with atmospheric carbon dioxide. So: If it tastes good to you, then drink it.


  And also, when boiling the water we loose bicarbonates, so is boiled water unhealthier?


When you boil water, you are increasing the concentration of dissolved species that do not boil off at that temperature. You can, for example, increase levels of nitrate and nitrite in water by boiling it, which is a bad idea if your water is contaminated by large amounts of nitrate/nitrite. Conversely, you can kill some if not all harmful bacteria in water by boiling it. And I don't think that boiling water results in the loss of bicarbonate in any measurable way, at least not on timescales we're interested in (like cooking).