Crush is the busiest time of the year for wineries and Winemakers. The tempo of crush tends to be dictated by the weather, something totally out of our control.
What we can control is how we treat the juice and must once the grapes are crushed.
Harvest additions are dependent on the conditions of the grapes when they are harvested, the stylistic goals for the wine and the personal style of the Winemaker.
Why add yeast?
First, you don't need to add yeast. Most people add yeast because they believe their favorite strain will give them better flavors than otherwise, or that their preferred strain won't be as likely to stick.
If you believe your yeast strain is important to your wine's character, then add that yeast. If you want to save a little money and want to tell customers and wine writers that you practice native fermentations (you modern-day cowboy, you)... then you might let nature take its course and not add yeast.
But why is yeast important? Because it converts sugar to alcohol, and that is a good thing. And grapes are covered with yeast (good and bad strains), and your equipment is probably covered in yeast (good and bad strains, even though you cleaned it.)
Even if you're a native yeast fermentor, you should be familiar with yeast rehydration enough to fix any stuck lots you might have (and have the name of a strong bayanus strain to call in case of emergency.)
Why add SO2 at crush?
Grapes are covered in wild yeast and bacteria, most of which would render grape juice unpalatable if left to do their thing. Thankfully most of these strains don't like SO2, so you can do your wine a favor and and some SO2 to the crushed grapes to inhibit these strains.
The wild yeast that you want to survive (assuming you're a daring soul who does not inoculate) can survive in the presence of some SO2 and not be bothered.
SO2 can also protect juice from some enzymatic browning (due to polyphenol oxidase for you enzyme geeks, you know who you are). Note that SO2 has no effect against the browning enzyme Laccase (produced by the Botrytis mold.)
Why adjust TA at crush?
If you live in a warm climate, chances are your grapes develop ripe flavors late enough that the grape's natural acid has been diminished. In this case you should adjust the acid back to what you, the Winemaker, think is the appropriate level.
Like with most things, the earlier you can solve a problem, the better. Adjusting the acid before or during fermentation is better than waiting until after fermentation or ML conversion to do so.
Even if you still need to make an acid add later, it would most likely be a small acid correction, as opposed to a major change in acid and pH.
Why adjust Brix?
Brix is a measure of soluble solids, and in grape juice that is mostly sugar. Our friends the yeast convert this sugar into alcohol, and the resulting alcohol plays an important role in the character and balance of our wine.
Depending on where you live, grapes are probably ripening with sugars higher than you want, or lower than you want (and rarely just where you want). In these cases it's important to correct the sugar content of the juice or must so that you end up with an alcohol content that will fit your wine stylistically.
In warm climates that means adding water to the must to reduce the sugar content, and in cold climates that means adding sugar to the must to increase the sugar content. Note that you can add either granular sugar or grape juice concentrate to achieve this goal, and depending on where you live one of these options is probably illegal (there, you've been warned.)
If you're dealing with a variety of grapes that tends to raisin (such as Zinfandel) or a vintage with a higher percentage of raisins, you should wait 2-3 days for the must to 'soak up' the sugar out of these raisins before correcting the Brix.
Why add nutrients?
Nutrients are required by yeast to ferment sugar into alcohol. The primary yeast nutrient is ammonia (NH3), added as Diammonium Phosphate (DAP).
There are lots of other amino acids, minerals and other so-called micro-nutrients required by yeast that may or may not be present in the juice. In this addition calculator they are referred to as "Generic", since you're adding a rate of the overall mixture, not a particular component of the mixture.
If yeast don't have enough of a particular nutrient, the fermentation will likely stop (or stick, as it's called), so you want to make sure you give the yeast the food they need to do their thing. Adding excess nutrient, however, can leave behind enough nutrients to encourage unwanted microbial growth. Damned if you do, damned if you don't.
A good rule of thumb is higher nutrient levels for higher brix levels, and also higher nutrient levels for moldy fruit.
It's not you're fault the fermentation stuck, so don't feel bad. But you're probably responsible for fixing it, so get to work!
Wine with residual sugar is an invitation for spoilage organisms and a headache to have in the cellar, so it's better to referment and get the wine dry than deal with the alternative.
It's best to understand why the fermentation stuck to know the best route to referment. If you're not sure why, take a sample to an outside lab and ask for their advice.
The procedure presented here is not by any means the best or only option, it's just one that I've found works consistently over the years. Pay attention to detail, though, as timing and dosage are important for the referment to have success.