​ We can find many influences of Spanish rule in the Americas lasting into post-colonial times. I've written before on colonial administration and on how certain practices were established there that facilitated later corruption - I'm adding the answer below as it seems relevant. Other factors include: the division into political units (incl. Viceroyalties and audiencias) would shape how independent nations in Latin America formed. And the Spanish monopoly for all American trade over Sevilla/Cádiz, which meant little contact and possibilites for cooperation between the different regions in colonial times. This conncects to a colonial economy primarily built on extraction from resources in Spanish Americas for transport to Spain. This meant a Spanish American economy where resources where not processed in situ but overseas, which continues in many ways and has negative effects on Latin American economies. I should add here that Madrid's direct political power was limited for example through the large distances for communication, the colonial and local administrations involved as intermediaries etc.

To this colonial influence I'd add that there are many other factors to take into account - it's too easy to simply blame current economic/political problems only on colonial administration. For example into the 19th c., Spanish control was strongest in central areas, but in many more peripheral regions there was little to no control over native societies. So that there is a strong indigenous influence (how strong always depending on which region), which continues to struggle with Spanish and other influences in complex ways. Another factor that's often neglected is that actually many more Africans (for the most part slaves) were brought to the Americas than Europeans emigrated there - at least until the 19th c. Lastly, the newly independent nations in the early 19th c. built on colonial precedents, but also on contemporaneous European and North American political ideas. E.g. the French and American revolutions were often quoted in different contexts. I'm talking about some of these later points in more detail in this older post which focuses more on Bolívar, esp. in the second and third parts.

Here comes the answer on colonial administration and corruption in Spanish America and Mexico, with further questions on these let me know.

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I. Colonial Spanish America

The development of colonial Spanish America during the 17th century was influenced especially by the Spanish "motherland's" internal and external crises. These led to the loss of the SPanish hegemonial position in Europe, economic and demographic decline and socio-political difficulties. Spain's crisis strongly influenced its overseas possessions: Signs include the continuing decline of the native population, and countless regional and local unrests which led to deep transformations of economic structure. Before this context we can see the SPanish Crown's increasing need for finding new sources of revenue.

Already under Philipp II. started a primacy of the financial politics which led to the introduction of new taxes and to voluntary donations. What is more, the Crown started giving out privileges and even the highest positions as well as noble status in return for "financial donations". This meant a major change from earlier positions of the colonial policies. Through more and more concessions the Crown tried to buy the acceptance of the colonial elites. The authorities tolerated misrule against the policies protecting native people, as well as various forms of corruption, and legalized illegal usurpations of lands. Over the long term this attitude led to a decrease in official authority, an increasing autonomy of the colonial administration, and thus to a strengthening of the colonial oligarchy's power.

While the purchase or sale of offices was already started during Philipp II's reign, at that point it was still limited to the lower administrative hierarchies. For sale were mostly notary's offices, positions in the city councils and in police functions (alguacilazgos). These positions did not for the most part include judicial or political-administrative functions, and were not tied to a fixed income. While they did have important duties, their sale did not yet mean a transformation of the administrative system.

This changed in the 17th century, especially under Philipp IV. and Charles II. who expanded the sale of offices massively - by now even the highest offices were filled by the Crown as a reward for payments. An official sale was judicially not possible, and so these offices were given out "in recognition for financial services to the Crown". The persion did not receive the office as their property, but only according to its usual conditions. Then again, de facto this did come very close to a sale. It also led to people just delegating their position to others - eg. high positions such as general captaincies and judges at the audiencias. The poisitions' high prices also meant that, quite clearly, people exptected to profit from them. Personal qualities were no longer essential for attaining certain key positions. Moreover, increasingly "favorites" of the Crown were appointed to viceroy positions overseas, which meant that the important impulses of earlier viceroys were now lacking. Horst Pietschmann (on whom I'm mostly drawing for this) adds that on the other hand, the sale of offices also meant increasing power for creole elites; they had been mostly excluded from higher positions before.

Pietschmann notes that during the second half of the 18th century this practice of purchases of offices was largely abolished. After 1750 it was mostly restricted once more to lower positions. On the other hand, while the Bourbon Reforms of the 18th century led to major policy changes, he also highlights the continuing incluence of traditional measures during this time. And moving ahead towards the independence period, we can also note how creole elites would take up power in many regions - elites which (as mentioned) had profited from and were used to the colonial sale of offices.

Moving towards the 2nd part of your question, I'll also briefly address the question of corruption in this context. Christoph Rosenmüller distniguishes colonial ideas of corruption from current ones:

Many practices that are considered corrupt today were culturally acceptable then, such as providing relatives and friends with jobs in tax collection, notarial offices, or the court. Renaissance thinkers even considered nepotism a pious duty, and the social networks that grew from such transactions in many ways wove together the fabric of society. [...] Most people in the Spanish empire did not consider nepotism within reason as offensive, but they did object to abusive judges. Corruption (corruptela) in the imperial sense typically meant that judges had violated justice by obtaining their posts inapprorpiately, accepting bribes, forging documents, or extorting the public.

[...] These specific semantics of the word corruption began drifting in the 18th century. The royal administration grew, and more functionaries who wee not judges collected taxes and enforced rules. Slowly the idea of corruption expanded beyond the judiciary to all branches of governance. In 1751 the viceroy of imperial Mexico ... lamented to the chief clerk of the COuncil of the Indies about not being able to "correct those things that happen in abuse and corruption", referring only to tax collectors and not judges. Up to that point, the fraud and excesses of nonjudicial officials had been a seperate matter from corruption; now they became part of the same idea.

Bringing this point back to the sale of offices, we can see a change in attituted towards such practices: From certain practices being considered non-problematic which we would today see as corruption, to others, including the sale of offices being increasingly seen as corruption since the 1750s - when this practice was first abolishes.