I wrote an answer to a very similiar question in the Colonialism AMA, dealing with how profitable the colonies were for the metropolis. My focus is here on the Spanish empire and its economic effects on other European powers, which I'm more familiar with than with later empires - seeing how your question is very broad. I hope it's ok to quote the earlier answer here, and I can add to it if there are questions.

Spanish America:

In the Spanish colonies, as in other empires, the profitability varied over time and depended on various factors. I'll first turn to economical and then to a few political factors.

Together with its constant warfare inside and outside of Europe, the huge influx of silver from its American colonies has actually been described as another factor in the Spanish empire's eventual decline. This argument holds that the silver profited bankers from other European countries (including the later Netherlands, as well as Genoese and Germans), as the Spanish monarch and elite borrowed heavily from them. The system of borrowing kickstarted by the Spanish overseas possessions can thus more generally be seen as one pillar of European imperial expansion.

John Elliot (in “The Old World and the New”) describes changes in Spanish economy over a longer period between the 16th and 17th cs. For Charles V. in the mid-16th c. he still sees his empire as a largely European one, as his sources of power remained mainly European – thus between 1521 and '44 the mines of the Hapsburgs produced nearly four times the amount of silver compared to the American ones. This started changing after 1550. Nonetheless, over the years American payments amounted to yearly roughly 250.000 ducats, not enough to compensate for the dangerous lowering of money value due the decline of traditional sources of income. Over time, inflation in connection with the large silver amounts proved to be another difficulty. This was reinforced by difficulties of levying taxes in Spain itself due to lacking centralised administration in 16th c. Spain.

Under Charles' successor in Spain, Philip II. the transatlantic trade focused on the monopoly of Sevilla – his empie became more clearly an Atlantic one, although the main income still came from Castile and Italy. With rising profitability, the “West Indies'” revenue made up 20 to 25% of Philips' income towards the end of his rule: For Elliot, the silver kept the imperial machinery working.

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In the second part I focus more on how these economical factors influenced (esp. Spanish) political legitimization of empire:

Turning now towards politics, we can see already in the mid to late 16th c. the influence of both the Spanish perceived riches and its hegemony at the time on its European neighbours. Both France and England started (first without much success) intervening stronger in the Americas – first simply in order to damage the Spanish standing there, which included the use of piracy. While the Spanish had justified empire partly with its unique territorial expansion under Philip and partly with their providential mission to conquer, the other powers turned to other mechanisms of justification, including the supposed rights to “uninhabited” lands (see Pagden's “Lords of All the World” for more details on this).

Apart from the critiques of its imperial rivals, criticism in Spain itself increased during the 16th c., and in the 17th c. Suárez de Figueroa went so far as to describe Spain as “the West Indies” of the Genoese to whom it was heavily indebted at the time. Other critics lamented a lack of trade with neighbouring countries instead of the Americas. Lastly, 1639/40 can be seen as an important turning point, with financial distress due to the war with Spain leading to continued interventions by the count-duke Olivares in the trade of Sevilla, in this way heavily damaging American trade at the time -- which in turn aided English, French and Dutch colonisation in the Carribean, by then clearly breaking Spains' imperial monopoly.

Due to such complex developments (of which I could just provide an overview here) it's hard to determine exactly how profitable the Spanish colonies proved in the 16. and early 17. cs. On the one hand they surely provided the means for further expansion and consolidation of royal power. On the other hand they played in the hands of the other European powers, both through Spanish reliance on foreign bankers and through the negative view of Spain related (in part) to its hegemony at the time.

Due to the question's focus on empire I looked at Spain's benefits here. Important consequences of these Spanish profits included the exploitation of native workers in mines (like the infamous Potosí, then in Peru), and more generally the large-scale appropriation of traditional native lands throughout Spanish America. For one example of this latter development I wrote an earlier answer on land rights in colonial Mexico.

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European context/ Summing up:

To go back to your broader question of the influence of the American colonies on Europe, I'd highlight two points, one economic and one political. 1) Spain's overseas possessions were of lesser importance to the empire in the earlier phases of Charles' V rule compared to the European possessions. However, over time they would be of major importance for Spain to finance its debts to merchants and banks of other European countries; and specifically to finance the near-continuous state of war the Spain maintained under Philipp II. You mentioned the Armadas - the cost of outfitting and maintaining them was also extremely high and lessened the profit margin.

Maybe more importantly, the huge influx of silver from the Peruvian and Mexican mines would profit rather other European powers over time. The Spanish crown owed huge amounts to German, Italian and later Dutch banker at various points in time. This increased the bankers' and merchants' influence at court. It also meant that in this way the American silver would provide a basis for the colonial ventures of later colonial powers, like the Dutch and British. Similarly, many profits from Brazil would go from Portugal straight to Britain, and help to finance British expansion. Last but not least, American silver traveled to Asia (e.g. from the Spanish Philippines to China) and fuel European-Asian trade and European colonisation in Asia.

Regarding 2): Another American influence was the development of discourses of how to legitimize colonisation. Spain under Philipp II. highlighted its unprecedented expansion (as "the realm where the sun never settle"s); but religious legitmization via the Christianization of native populations remained at least equally important. In response to Spain's temporary hegemony, other powers like Britain and France would focus more on the legitimacy of occupying supposedly "unoccupied" lands. What is more, during the whole colonial period we can see varying discourses on how to judge native populations in relation to Europeans. This could range from the native as "Barbarian" and "underdeveloped" to later, 18th c. ideas of a "noble savage". Such ideas are closely related to legitimization of colonial occupation, which is another topic in its own right.

Edit: Added context