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Generally speaking, I would say that the official term used in Spain was „Las Indias“ (the Indies) following Columbus' expeditions. So although you're right that the term „America“ was known and sometimes used in Spain – as it was the common term used in most other European countries-- , Las Indias had already found prominence as a term before America's coinage in the Waldseemüller map (discussed below) in 1507. By at the latest the mid 16th Las Indias was routinely used for colonial Spanish insitutions, in administrative sources and (even earlier) in chronicles aka Cronicas de Indias.

Although it's difficult to seperate both questions I'll first look more generally at the term America and similar terms in 16th century Europe, and from there move to the Las Indias concept in early modern Spain.

America/ Nuevo mundo

Edmundo O'Gorman has traced the concept of Columbus having found a new part of the world. He famously argued that it is not possible to "discover" something ex nihilo that was already there. For this retroactive process of describing America as a continent only "waiting to be discovered" he coined the term "invention of America". O'Gorman traced the de development of such ideas from Columbus through various writers influenced by him, including Las Casas and Columbus' son, to become highly influential on later perceptions of America.

While interesting conceptually, your question deals more with the term America and its use in Europe and/or Spain. I've already mentioned the official term "Las Indias" in Spain, but of course there was a multitude of other terms floating around in Europe and Spain in the 16th century, often with no clearly fixed meaning - other examples include "Nuevo Mundo" (New World), following Amerigo Vespucci, and "The Fourth Part of the World". Specifically for "America", we know the term was introduced by Martin Waldseemüller in his famous map of 1507 (map link - it has "America" written on today's South America).The map built on Vespucci's name and a letter from 1503, now known as "Mundus novus", in which he argued that the land mass spanning to the south of Cuba was not the Asian coast but an unknown part of the world - crucially he called it "new" instead of "unknown", contributing to the "invention of America" mentioned above. Although Waldseemüller would later change his view of the newness of America, the damage had been done so to speak. Sebastian Sobecki sums up some of the following developments and terms in use:

The term gained momentum only during the second half of the sixteenth century, following Gerard Mercator’s (1512–1594) projections of 1538. The spatial and narrative invention of the American continents remained an unfinished project well into the seventeenth century, despite the circumnavigations of Ferdinand Magellan (c.1480–1521) in 1519–1522 and Francis Drake (1540–96) in 1577–1580. Spain did not adopt the name “America” until 1758, and English writers, usually drawing on inferior, homemade cartography, continued to struggle with the name and the newness of the continents. John Dee (1527–1609), self-appointed cartographer and imperial ideologue to Elizabeth I, insisted as late as 1580 on the term “Atlantis” for the Americas, and in 1595, Walter Raleigh (1554–1618), having trusted Dee’s maps and advice during earlier attempts to colonize Virginia, went searching for El Dorado, a bizarre conflation of distorted Spanish reports of Muisca traditions and faint echoes of India’s famed gold.

We can note here that not only in Spain but also in England (and presumably other European countries) there were various terms for what we know usually call America, and that their definitions were often far from clear and could incorporate mythical elements. Speaking on the "New Found Land" (nicely ambiguous term), Sobecki notes that in England prior to English colonial endeavours, it was often seen as a

no-place, not an exotic world but a forlorn outpost in the cold North Atlantic. These works do not testify to any noticeable interest in the New World or, for that matter, new worlds; they notice the latest additions to speculative geography in passing and with indifference.

While specific to England, we can see similarly how in the early 16th century Spain a lack of knowledge of the Americas would have led to confusion regarding terminology. Only towards the mid-16th century do we find detailed maps depicting the Americas, such as Sebastian Münster's Novus orbis. According to Walter Mignolo "It was toward 1555 that the world began to look to our hypothetical European observer very much as it does today for many people on this planet." This knowledge included Spain, where Charles V had given the quite accurate Agnese map of ca. 1546 to his son Philipp II, who by the mid 16th century ruled over the Spanish overseas territories (here's a link to the map by Battista Agnese, which looks certainly more accurate than Waldseemüller's earlier one). But while the term "America" can be found in various Spanish literary and cartographic works before the mid-18th century, it did not hold an equal status to the "official" Las Indias.

Interestingly, the term America and especially the form „americanos“ (Americans“) had become important by then in Spanish America - even before the time when the term "Las Indias" was officially abolished. Leading up to and esp. during the wars of independence of the early 19th century, creole elites would refer to themselves as „americanos“ as a way to distinguish themselves from the Spaniards (often called „gachupines“ pejoritively). So at least by this time, I would say that the term americanos resonated more in Spanish America than in the Spanish metropolis. This would only increase with the French coinage of „Latin America“, as a tactic to highlight French influence in the region.

Not to make this even more confusing, but in my readings of native colonial Mexican authors (16th/17th centuries), the term „Nuevo Mundo“ also comes up sometimes. This would have been taken from European authors of course, but was used by some of these Mexican authors to refer to „this New World“, in order to show that the pre-colonial history of „their“ continent (sometimes meaning just Mexico) was in many ways equal to the „Old World“. As I mentioned, the term New World goes back to Vespucci's notions of having "discovered" a heretofore unknown part of the world - e.g. Waldseemüller similarly talks of a Terra Nova in this context.

Las Indias

The judicial situation of Spain's overseas possessions was however quite complex, and changed over time. It's even difficult to speak of colonies, as they were only designated so in later centuries. The Spanish Crown juristically denominated them as „Reinos castellanos de Indias“; the British Crown called them „Indias Occidentales“ to distinguish them from the „Oriental Indies“ meaning Asia. The Occidental Indies term was also sometimes taken up in Spain.

The influential Chronicler-Cosmographer of the Indies, Juan López de Velasco, defined the Indies so in the 1570s (translation from Mignolo, p. 270, cited below):

The Indies, the islands and terra firma in the Ocean which are commonly called the New World, are the lands and seas which lie within the boundaries of the Kingdom of Castile, which is a hemisphere, or half of the world, beginning at 180 degrees west from a meridian circle which passes through 39 degrees longitude west of the meridian of Toleda.

By this time the Indies had officially replaced earlier, more vague terms including "New World" (mentioned above). So the „ Castilian kingdoms of the Indies“ were then recognized as kingdoms distinct from the Crown of Castille, as was the case with other European kingdoms associated with Castille. This complicated construct meant that the „kingdoms of the Indies“ pertained to the Crown of Castille, but not to the kingdom of Castille – the kingdom of Castille was tranformed into an administrative hierarchy that was superior to but did not hold dominion over the Indies. However, the Indies did have their own law code („derecho indiano“) based on Castilian law. This special position led the overseas territories having their own administrative organs (with the „Consejo de Indias“ at the top), which you mentioned, with the Consejo being responsible for all matters save foreign policies.

To make this a bit clearer maybe, I should note that the Catholic Kings (Ferdinand and Isabella) at first took the Indies to be their personal patrimony. Under Chales I. they formed part of the „realengo“ (royal possessions), and individable from the kingdom of Castille – they did not have lesser rights, and thus formed an integral part of the Castilian monarchy. During the late 17th century parts of the Indies even received their own kingdom status, which influenced forms of patriotism there.

All this changed in the second half of the 18th century, when the Indies were declared to be „Colonias“ or colonies under Charles III. With Napoleon's invasion of Spain, and finally with the Spanish constitution of 1812 the term „ Castilian kingdoms of the Indies“ was abolished and they were incorporated into the „Reino de las Espanhas“ – reflecting the inner-Spanish political crisis and in connection the Latin american wars of independence of the time.

Edit: Added 2nd part