Strong acids completely dissociate in water and donate all protons. In a weak acid, only a fraction of the acid molecules dissociate.
The double arrow depicts that the reaction goes in both directions until equilibrium is reached. At that point, the forward reaction occurs at the same rate as the reverse reaction. For strong acids and bases, that reverse reaction is almost non-existent. This is why only one-directional arrows are used.
The strength of an acid is represented by the acid dissociation constant, Ka.
Whether an acid is strong or weak has nothing to do with the concentration or dilution of the solution. For example, a strong acid like sulfuric acid would not become a weak acid just by being diluted. Although its pH would be raised, the acid would be less concentrated, and the solution would appear less reactive, it would still be a dilute solution of a strong acid.
Be careful when working with concentrated strong acids!
The reaction of strong acids with water can be very dangerous. The dissociated hydrogen ions react with water molecules to form hydronium ions, and this reaction releases a lot of energy. This exothermic reaction can be very strong and may cause the acid to boil and spatter.
The safest method is to add the acid incrementally to the water, rather than the other way round. This way, a dilute solution of strong acid is formed first, and the exothermic reaction can be controlled.
Acids are corrosive! Learn how to handle corrosive chemicals.