If A reacts with B (aq), does that mean it is soluble in it?
Summary, TL;DR:
No, A reacting with B does not mean that A is soluble in B. Solvation and chemical reaction are two different processes, although the distinction can in some cases seem less than perfectly clear. All of the example compounds you discussed are soluble or miscible with both water and hydrochloric acid. The IUPAC definition of a chemical reaction is "A process that results in the interconversion of chemical species", which is fundamentally different from solvation, and does not apply to any of the examples in the question.  



As you suggest, solvation and chemical reactions are two fundamentally different concepts. For example, if compound A reacts with water to form a homogeneous liquid solution, that is a different processes than if compound A were to simply be dissolved by water.  

Solvation is defined by the IUPAC Gold Book as follows:  


  Any stabilizing interaction of a solute (or solute moiety) and the solvent or a similar interaction of solvent with groups of an insoluble material (i.e. the ionic groups of an ion-exchange resin). Such interactions generally involve electrostatic forces and van der Waals forces, as well as chemically more specific effects such as hydrogen bond formation.


Whereas the IUPAC Gold Book defines a chemical reaction as:  


  A process that results in the interconversion of chemical species.  


One aspect of solvation vs. reaction that may seem confusing is the solubility of ionic species. Even though $\ce{NaCl}$ becomes $\ce{Na+}$ and $\ce{Cl-}$ in an aqueous solution, this does not constitute a chemical reaction as defined above, and we say that $\ce{NaCl}$ is soluble in water. The same is true of your example of $\ce{H2SO4}$; even though it dissociates in water, it is not converted to a new chemical compound. One way to think of this is if you remove the solvent, the solute should typically resume to it's original form. From our previous example, if we evaporate the water from our aqueous solution of $\ce{Na+}$ and $\ce{Cl-}$, we just get the solid $\ce{NaCl}$ back.  

Your examples of solubility in hydrochloric acid are a bit complex because that is a two-component system of water and $\ce{HCl}$. All of the compounds you discuss are water soluble and it doesn't matter if the $\ce{HCl}$ is there or not. One slight exception in your examples is ethylamine. Ethylamine itself is miscible with water, but many higher molecular weight amines are not water soluble, but are soluble in hydrochloric acid. In this case, the $\ce{H+}$ from $\ce{HCl}$ protonates the amine, making the hydrochloride salt. This is still an example of solubility however, as once you remove the solvent, you are left with the original compound, in this case the amine.