Can the human body store protein?
The human body can not store Proteins (technically).
BACKGROUND :
PROTEIN is a very broad term and there are hundreds and thousands of proteins.[1]
Proteins are heteropolymers consisting of amino acids held by peptide bonds.
Amino acids :
There are 9 amino acids which we need to intake.

Phenylalanine, valine, threonine, tryptophan, methionine, leucine, isoleucine, lysine, and histidine.

Amino acids our bodies can produce:

These five are alanine, aspartic acid, asparagine, glutamic acid and serine.

These amino acids combine in different arrangements to give those thousands of Proteins. This process is carried out by Ribosomes (cook) and mRNA (recipe) inside every* cell.
Protein metabolism in humans :
During human digestion, proteins are broken down in the stomach to smaller polypeptide chains via hydrochloric acid and proteaseactions. This is crucial for the absorption of the essential amino acids that cannot be biosynthesized by the body.
Trypsin and Chymotrypsin as Trypsin and Chymotrypsinogen are the enzymes  necessary.
.
MAIN ANSWER  :
The excess Amino Acids are broken down into Glucose ( carbohydrate like stuff ) and the Nitrogen moiety is converted into Urea / Uric acids. This is then excreted by kidneys.

Amino acids can be broken down into precursors for glycolysis or the Krebs cycle. 
Amino acids can also be used as a source of energy, especially in times of starvation. Because the processing of amino acids results in the creation of metabolic intermediates, including pyruvate, acetyl CoA, acetoacyl CoA and so on; amino acids can serve as a source of energy production through the Krebs cycle.

Therefore excess protein is simply used to make energy like carbs and the Nitrogen part is excreted. Now the excretion process can strain the kidneys due to increased work load, and prolonged stress can lead to malfunctions.
Also,
Our body is very concerned with conserving nitrogen levels, since our body can't fix nitrogen gas (N2) into ammonia (NH3). but, we can't just store huge amounts of ammonia because it's toxic. sugars and fats aren't toxic, so we can store them. because of this, our body either quickly utilizes excess ammonia to make nitrogenous compounds like amino acids and nucleotides, or it excretes it via Urea in our urine.
Problem with Protein storage :
If our bodies had to store proteins, it would need several sophisticated mechanisms and specialised storage cells.( Like we have adipose tissues short fats and lipids storage ).
if you examine the structure of amino acids, they are very similar to many of the carbon-hydrogen-oxygen metabolic intermediates. for example, alanine is an amidated pyruvate, glutamine is an amidated alpha-ketoglutarate, etc. many are easy to make, and most don't cost much to do so, so the body would rather use these intermediates to either:

Create acetyl-CoA, NADH, FADH2, and ATP for energy. this would happen during exercise or starvation.

Create glycogen, fatty acids, nucleotides, NADPH, and other metabolic compounds. this would occur after a big meal, during sleep, and when energy is not needed in high amounts. these compounds are very expensive ATP-wise.


So there's no need to store whole Amino Acids because they can be immediately used for whatever the body needs and then remade when necessary.
We do not store amino acids because we don't need to.
Right dose of Proteins :
Research shows:

A total of 80 g of whey protein was ingested in one of the following three conditions: 8 servings of 10 g every 1.5 h; 4 servings of 20 g every 3 h; or 2 servings of 40 g every 6 h. Results showed that MPS was greatest in those who consumed 4 servings of 20 g of protein, suggesting no additional benefit, and actually a lower rise in MPS when consuming the higher dosage (40 g) under the conditions imposed in the study. 

Check here for the interesting results ( this is very relevant to your second question )
Therefore distributed intake of right amount of protein is important!
Scientific literatures :
Proteins:
https://en.wikipedia.org/wiki/Protein_(nutrient)
https://en.wikipedia.org/wiki/Amino_acid
Protein Metabolism:
https://opentextbc.ca/anatomyandphysiology/chapter/24-4-protein-metabolism/
Protein intake and storage:
https://newsnetwork.mayoclinic.org/discussion/are-you-getting-too-much-protein/
https://www.self.com/story/what-protein-does-in-your-body