Enthalpy of neutralisation for weak acids and weak bases at standard state
Q1, well you are correct to a certain level. standard state of a solute in a solution of any specified solvent is the state having solution under pressure exactly equal to standard pressure 1 bar,composition of solute equal to 'a standard composition', solution must be an 'ideal solution (perfect solution)', and finally solution must show 'properties of an infinitely diluted solution'. here standard composition refers to either, standard amount concentration (1 mol/dm3), standard molality (1 mol/kg) or standard mole fraction (1). note that temperature is not a part of definition of standard state. this is a hypothetical state for any temperature we use or what standard composition you may chose. here you have chosen standard molality,its ok.(note that we can prove that for an ideal solution showing properties of infinite dilution at a given pressure and temperature molar enthalpy of solute is a constant what ever its composition may be.plug into somewhere that defines you exactly the terms 'perfect solution',infinite dilution and properties of infinite dilution to understand this, its easy). infinite dilution means amount concentration or equivalently any other composition term of solute approaching zero. however acid  composition is molality =1mol/kg as you prefer or any other standard composition, but yet it is not infinitely diluted although it shows properties of infinite dilution. key thing here to understand is that infinite dilution is one thing and showing properties of infinite dilution is another. this is not the only hypothetical state you are dealing with, standard state of a gas existing in pure form , as a solute or in a mixture is the pure form of gas at pressure 1 bar and gas is ideal or equivalently compressibility factor =1. going back to your question, we can talk about strong and weak acids or bases. in fact Ka and Kb values are based on this.  Q2; strong and weak are relative terms. the better word is strength. even strong acids or bases absorbs heat .