This is a great question but also a difficult one to answer because it is so varied. The Spanish Empire was really an empire of paper.
Documents, letters...everything really...was copied again and again and again. Often, the original would be stored in a local archive, a copy sent to another party or parties, another copy sent across the ocean. Someone would put it on a desk, and another document on top of that, and another on top of that, and so on until eventually, the pile was bound together and put on a shelf somewhere, where it sat until someone needed to consult it, or until someone forgot about it, or until someone decided to collect them and preserve them somewhere, sometimes by taking them out of the town or country where they originated. Thus, Spanish documents are scattered in archives, libraries, and private collections across the Americas, Spain, Portugal, and Europe. This includes national archives in nation-state capitals, regional archives, parish archives, notary archives, museums, military archives, and university archives just to name a few. Some of these archives are still functioning archives. Their documents are still consulted by the general public in land disputes or for genealogy at the same time that historians are doing research for their books and articles!

Most colonial history publications are based on a combination of materials from local archives and the Archivo General de Indias, which is the main imperial collection of colonial documents that were sent across the Atlantic (and Pacific sometimes).

Every one of these archives store their documents slightly differently depending on local climates and local financial situations. I am not an archivist, so I have no training in specific preservation techniques, but basically, most modern archives have a climate controlled environment to store the documents, which keeps low humidity. Archivists and preservationists attempt to keep the documents pest-free and mold free, while protecting them from water, light, or heat. Many places even carry out restoration and digitization of damaged documents. For instance, you can access the Spanish archives using this link (http://pares.mcu.es/). The website is terrible, but run a search for something (like "Drake") and check out what colonial documents really look like. (Just always use the yellow atras button and not the back button on your browser.)

Poor preservation is common in poorer nations, cities, and parishes. For example, I have worked in archives in the tropics that are not climate controlled at all. This caused mold to grow on the pages. Many of the pages have evident water damage. Sometimes, the ink that they used during the seventeenth and eighteenth century was acidic so it ate through the page. When I turned the page, the empty part of o's and e' and a's and d's would fall out of their places on the page. It was so sad.

Also, from what I can gather, there is some debate between archivists on whether a researcher should wear gloves or not when working with the physical documents. Some archives require that you wear gloves and a mask to protect the documents from the oils on your skin and the humidity of your breath. Others say that wearing gloves is bad because the loss of dexterity means you can't gently turn the pages, which causes far more damage than the oils on the hand. For this reason, many archives and libraries don't have you wear gloves any more.

Of course, natural disasters, fires, accidents, and civil wars have destroyed priceless documents, collections, and archives. For example, hurricanes in the Gulf of Mexico have flooded archives. When Hurricane Maria that devastated Puerto Rico, many libraries and archives across the island were flooded, and without power or emergency preservation techniques, many of the documents have been destroyed. Another problem is that documents get stolen. A few times, I have called a document, and it wasn't in the folder it was supposed to be in. I talked to the archivist and they were like...Well, it's gone I guess. Most wealthy archives and libraries I've worked in have elaborate systems of security to keep researchers from sneaking out documents.

Some archives are better organized than others. Most have some form of searchable catalogues that help researchers find documents. Sometimes these include lengthy descriptions of documents, but mostly, they are just vague descriptions that tell you roughly what a volume or box contains and the approximate years of creation. Or maybe they only tell you what the collection is named or what part of the government created the document. It is in these masses of uncatalogued collections that some of the most amazing documents are found by sheer chance. While every researcher dreams of coming across a field-changing document like Guáman Poma, the vast majority of documents in archives are mundane documents generated in the day-to-day operations of colonial government. These include land titles, bills of sale, lawsuits, letters, marriage records, church records, military records. Individually, they tell historians almost nothing, but collectively, they can tell us a great deal. For the most part, that is how history is written these days: through slow, tedious collection of mundane data from every day documents.