Assumptions Built in to Solving Titration Equilibria?

[QuestionMarks]

I'm trying to reach a more thorough understanding of what's going on when we calculate equilibrium concentrations and was wanting to understand more what assumptions we make in order to follow through with the calculations.

Let me ask off of an example:
Consider mixing equal amounts of NH3 and HCl, each having the concentration of α upon mixing.

The general way I've seen taught is to assume via being a strong acid that the HCl fully ionizes and causes the following equation to occur:
NH3 + H30⁺ → NH4⁺ + H2O
Then to recognize that NH4⁺ is a weak acid, so we'll need to set up the reverse equation as an equilibrium to run an ICE problem off of:
NH4⁺ + H2O ←→ NH3 + H30⁺
Wherein we take the initial concentration of NH4⁺ to be α, and can use it's K_a to find the final hydrogen ion concentration of the entire solution.

Here then are my questions directly:
1) In the first equation, we seem to assume that all the hydrogen ions from HCl jump onto our NH3, thus allowing us to use our initial α concentration also as the initial concentration of NH4⁺. How so? Could it not be that some hydrogen ions tend to stick around as H30⁺, thereby reducing the actual concentration of NH4⁺ to slightly less than α? What knowledge/assumptions do we use to reduce this complexity from our calculations?
2) Are there any other important assumptions we make when working similar equilibrium problems? As I work them myself, I constantly feel like the scenarios are a bit oversimplified (even if I acknowledge they give us answers very proximal to reality).

Thanks much!

 

[QuestionMarks]

Reasonable time for me to bump?

 

[Borek]

I strongly suggest you go through the http://www.chembuddy.com/?left=pH-calculation&right=toc

Not only it shows how to solve the general case of acid/base equilibrium without any simplifying assumptions, but it also shows what the simplifications are in practical working of many cases. Once you know what the general approach is you should be able to understand why and when simplifying assumptions work (and when and why they don't).