In a solution containing multiple species that undergo acid/base chemistry, everything will react with everything (including water, which can act as acid or as base). Equilibrium is established quickly, at some pH. Ingredients with a $\mathrm{p}K_\mathrm{a}$ higher than the pH will be (mostly) protonated, while ingredients with a $\mathrm{p}K_\mathrm{a}$ lower than the pH will be (mostly) deprotonated.
I am guessing that the indicator stock solution is formulated to be neutral initially, judged from the yellowish greenish color shown and described on the internet.
Why is it that the various components of the universal indicator are preserved in their neutral state?
Indicators don't have a neutral state. They have a protonated and a deprotonated state, one of which predominates at neutral pH (unless the $\mathrm{p}K_\mathrm{a}$ is near 7, in which case both states will be present). To make the pH neutral after mixing all the indicators, you would titrate the solution until it is neutral. In this process, some indicators will get deprotonated (and change color).
Methyl red is a weak acid, and sodium hydroxide is a strong base, so won't the methyl red in the universal indicator react with the sodium hydroxide?
Methyl red is red in the protonated form and yellow in the deprotonated form.
The $\mathrm{p}K_\mathrm{a}$ is 5.1, so if you titrate a solution to neutral with NaOH, you will have the deprotonated yellow form. In fact, the indicator substances were chosen so that they are either yellow or colorless at neutral pH. As you change the pH to acidic or basic, you get the "darker" colors like blue, red or purple.
Here is an image showing the colors of the four indicators (bottom) along with the color of the universal indicator, which is a mixture of the four. If you pick a pH, you can rationalize the color of the mixture. For example, for neutral pH, you have one indicator in the colorless form, two in the yellow form and one in the green form; this mixes to a greenish yellow. To get blue at pH 10, the concentrations of indicators have to be chosen carefully such that the yellow from methyl red does not overpower the blue from thymol blue and bromothymol blue.
Source: Adapted from https://www.compoundchem.com/2014/04/04/the-colours-chemistry-of-ph-indicators/
