In analysing amino acids content in a protein through gel electrophoresis, What's the purpose of the gel? Wouldn't putting the amino acid in the gel prohibit the amino acid from dissolving into the buffer solution (which its pH controls the protonation and de-protonation of the amino acid)?
In gel electrophoresis, the gel is the mechanism by which macromolecules of different sizes are separated. By loading the gel with amino acids (or proteins or DNA), you start all of the samples equally, and then push them through a gel using a salty buffer solution that is electrified.
The gel has a certain concentration of a polymer (commonly agarose or polyacrylamide) that acts as a mesh that slows your macromolecules down. So, after running a gel for a certain amount of time, molecules of different sizes will have traveled different distances. This can be visualized by different methods (ethidium bromide for DNA and antibodies for proteins, commonly) after further preparation.
So, the amino acids in question should not be dissolved in buffer, as this would result in loss of sample.
Regarding the last point: when loading samples on a gel, they must first be dissolved in some loading buffer that will help the sample settle into the well and not diffuse out. I took your mention of "buffer" in the original question to be the running buffer for electrophoresis, which you should not dissolve your sample in. I am also unfamiliar with analyzing amino acids by electrophoresis, and I do not know if there is a specialized loading buffer that kind of analysis would require. A quick look didn't uncover anything, but you might be in a better position to answer the loading buffer issue.