Change in Gibbs free energy vs. change in enthalpy
To start off, $\Delta G$ is concentration-dependent and $\Delta H$ is not. You already showed the concentration-dependence of $\Delta G$. There is no equivalent relationship for $\Delta H$ because it does not depend on concentration. $\Delta H$ reflects how the bond energies change from reactants to products. Bond energies don't depend on concentration. (It also reflects other more subtle things like solvation and intermolecular interactions, but these don't change in an ideal solution).  In contrast, $\Delta G$ has an entropy term, which has to do with the entropy of mixing, resulting in a significant concentration-dependence even in ideal solutions.


  Is this understanding correct? Or does ΔH change throughout a reaction like ΔG?


The enthalpy H of the system changes throughout the reaction. Because the enthalpy is not concentration-dependent, it changes in a linear way with the amount that reacts. There are two versions of $\Delta H$ with different dimensions and units. One does what you say: compare H between two states. It has dimensions of energy (typical units: kJ). The other asks what the change of enthalpy is per amount reacted. It has dimensions of energy per amount (typical units: kJ/mol) and is called molar change in enthalpy. 

An example is the molar enthalpy of formation. If you want to know how big the enthalpy change is to make a certain amount of a compound, you multiply the amount by the molar enthalpy of formation. The dimensions of the result will be energy again (units: kJ / mol * mol = kJ). Introductory textbooks sometimes gloss over the difference between molar (intensive) and extensive enthalpies.


  If ΔG gives information about a specific point in the reaction, then why is there a "Δ" at all?


The Gibbs energy G of the system changes throughout the reaction. The change per amount of product formed decreases as the reaction approaches equilibrium. Again, ΔG can mean two things, either the difference in Gibbs energy between an initial and a final state, or the molar change in Gibbs energy with given concentrations. 

You might say concentrations change while the reaction is going on, so it is impossible to make a statement about specific concentrations. You solve this problem by either looking at a very small amount of change (infinitesimal, if you want) or by imagining very large volumes of reaction mix, where concentrations change very little even if - say - one mole of reactants turn into one mole of products. If you want to communicate that this is what you are talking about, you can either use $\Delta_r G$ or $\frac{dG}{d\xi}$. Both have dimensions of energy per amount of substance (typical units kJ/mol). 

Again, introductory textbooks tend to gloss over this difference, not giving $\Delta G$ and $\Delta_r G$ different symbols even though they have different dimensions (one is extensive and the other intensive) and mean different things.