but this implies that somehow the quantity demanded is multiple values simultaneously at a single price point. How is this possible?
This just says that at that particular price people will demand any amount of goods between $[0,\infty]$.
There is no reason why the quantity demanded could not be multiple different numbers at a given price. That just means more than 1 quantity is consistent with price.
For example, in this graph, the quantity demanded at 4 dollars is somehow 10, 20, 30, etc. units all at once.
Yes, exactly. There is nothing wrong with that. That just says that at price 4 people could demand any number of goods. Of course only one combination will actually be realized in reality, but any quantity at that price is possible outcome.
Consider an analogy. A model for stock price could predict that stock price increase will be anything between 3-15%. Of course, eventually only 1 price change materializes but ex ante that one price change could be anything in the range. Similarly, with infinitely elastic demand at price equals 4 any quantity demanded will be consistent with demand but in real life only one quantity would materialize.
Also, when demand is perfectly elastic, the quantity demanded supposedly rises to “infinity” when the price drops below the equilibrium price (in this case, 4 dollars), but why doesn’t it show that on the graph?
It is shown in the graph, there is no way how you can graph an infinite line with finite piece of paper, but the fact that the line is not ended with some clearly marked point indicates that it goes on to infinity.