There's a different answer for every year of Napoleon's reign, but i'll take the year 1812, on the eve of the Invasion of Russia, as a good marker.

France proper: These are the departments of France, inherited directly from the revolutionary state built in the 1790's. At his coronation, the borders of the French state already included the Rhineland, Belgium and Piedmont. To this, Napoleon eventually added parts of Central Italy and Switzerland, the entirety of Holland, the entire North Sea coast of Germany along with Hanover, Dalmatia (the Adriatic coast of present-day Croatia), and even Catalonia. These were ruled by centrally appointed officials, and were therefore under the tight administrative control of the central government. Crucially for Napoleon, the departments contributed directly to the system of conscription, thereby filling the ranks of the Imperial Army.

Rhine Confederation: The Rheinbund was made up largely of willing partners in the Imperial project. Napoleon had enlarged the territories of the German princes at the expense of the Church and the Habsburgs, while elevating their titles and winning them independence from Austria. Napoleon was so confident in the support of the Germans that he rarely needed to garrison his supply lines when campaigning in Germany or further east. Bavaria, Baden, Wurttemberg and the other principalities provided contingents for Napoleon's military campaigns, usually not exceeding a corps in size. Nevertheless, Napoleon was forced to annex some of the northern states (including parts of his brother's Kingdom of Westphalia) due to their failure to enforce the Continental System.

Italy: The Kingdom of Italy - Lombardy, Venetia, and Romagna for the most part - was in personal union with France, since Napoleon was its monarch. Ruled by Napoleon's stepson Eugene as regent, the kingdom had its own army. Beyond this, the population of northern Italy was quite supportive of Napoleon, because he had expelled the Austrians from Lombardy and ended Papal rule in Romagna. The liberal and largely autonomous government faced little resistance and widespread support. Even today, Napoleon is remembered fondly in cities such as Milan. There was no need for a French garrison per se by 1812. Further south, in Naples, Napoleon had installed Joachim Murat as king, and French troops supported him in his futile quest to invade Sicily.

Spain: Over 200,000 troops were engaged in Spain before the Russian campaign. Joseph Bonaparte controlled almost all the cities, with the major exception of Cadiz. Spain was undoubtedly a war zone, with the guerrillas exacting a heavy toll on the French in the countryside, and the British sturdily checking any advance into Portugal. Moreover, there was no Napoleonic control over the Spanish colonial empire.

Prussia and Poland: Napoleon had totally defeated Prussia in 1806 and 1807. After carving out the Polish Duchy of Warsaw from Prussian territory, he maintained his supply lines to Poland by garrisoning all of Prussia. As a bonus, this allowed him to extract huge reparations from the Prussians and severely limit the size of the army. As a result, the Prussian state was utterly subservient to the French Emperor in 1812, being a part of the Continental System and required to contribute what soldiers it had left for the Russian campaign.

Sweden: The Swedes had chosen the French Marshal Bernadotte to succeed their ageing king. But Napoleon did not really gain an ally by this, since Bernadotte had very little loyalty to him and would switch to the Coalition's side at the first opportunity.

Austria: Napoleon had neutered the Austrians by splitting away some of the most populous parts of their empire. Nonetheless, he had been persuaded to leniency after the War of 1809, thanks in no small part to the promise of the Austrian emperor's daughter Marie Louise as his bride. Therefore, French troops did not occupy Austria as they did in Prussia. But the Austrians were forced into alliance with Napoleon, were brought into the Continental System and had to contribute men for the invasion of Russia.

The seas: In stark contrast to the army's freedom of movement on land, in all its adjoining waters the French were severely impeded. British mastery of the seas was complete. Seaborne trade had ground to a halt, and communication with the colonies broke down.

While this all makes a case for pretty thorough control by Napoleon over most of Europe, this outward show of power was deceptive. The Continental System illustrated the limits of Napoleon's authority. Essentially a blockade of all British trade while favouring French exports, Napoleon forced all the countries under his control, as well as Russia to subscribe to this war-winning scheme. However, not even his own brother ruling in Holland complied with the decrees, and throughout Europe, trade with Britain continued to flow. Exasperated, Napoleon resorted to annexing coastal territories and pouring troops into the port cities to enforce the System. Yet even in France, black market trade with Britain continued, and was in fact vital even for producing many of the French Army's uniforms. This displays the fact that Napoleon's military control over Europe did not easily translate into political control.

That became abundantly clear after the disastrous March on Moscow. With the French Army starved of troops and devoid of cavalry, the various subdued states quickly turned on Napoleon, with the Germans famously switching sides in the midst of the decisive Battle of Leipzig in 1813. Even men Napoleon himself had installed on their thrones, like Murat, eventually joined with his enemies. The French Empire, one of history's great European projects, was eventually proven to be a just a paper tiger. Without the army, it crumpled quickly.

TL;DR: Napoleon's armies had the run of Europe, but the Empire's vassal rulers were disloyal and blatantly skirted his commands.

Hello, I'm French and have always had a great passion for the history of l'Empire, which I have studied with great interest since I'm a kid. I will try to provide you with a little bit of insight on this matter. Sorry for eventual grammar mistakes, I do my best !

As you said in your post, Napoleon practiced what we call Nepotism, placing his family, closest and most loyal friends to high positions in Europe. For example, he placed his older brother, Joseph Bonaparte, who was previously King of Naples (placed in that position by Napoléon in 1806), as King of Spain in 1808. He then appointed one of his closest friend, the Maréchal Murat, as King of Naples, as a reward for his service.

Now, while all of Napoléon's direct family were titled 'Princes de l'Empire' and some were 'given' kingdoms, earning themselves the title of King (Joseph for instance), they all answered to Napoleon himself. In terms of actual power, they could be considered 'merely' Governers, taking care of the lesser matters that require a leader's attention.

Other allied states, such as the Duchy of Warsaw (Pretty much Poland), could be considered Satellite States who, although not ruled by a member of the Bonaparte family, still answered to Napoleon.

As for l'Empire's military strengh, it is estimated that, prior to the retreat from Russia in 1812, over a million soldiers served in the Grande Armée at a time (for an estimate of roughly 1.5 to 1.7 million over the course of Napoleon's reign).

The Army was organised in several "Corps d'Armée", each of one usually under the command of one of the "Maréchaux" (Many Maréchals de France died in Napoleon's campaigns, they were replaced at the head of each Corps by Generals, who may have been promoted to the dignity of Maréchal de France later by the Emperor)
Each of these Corps d'Armée consisted of several infantry divisions (each division having at its full strengh up to 15,000 soldiers), 2 to 3 cavalry brigades (several thousand soldiers) and artillery ( I do not recall the numbers on this one, I'll try one of my books later on and will eventually edit) Dividing the "Grande Armée" in such way allowed Napoleon to have extremely mobile armed forces (capable of traveling between 30 and 50 kilometers a day depending on terrain and weather), while allowing logistics to be easier to manage. L'Empereur could thus place and move these Armies around Europe wherever they were needed.

Now, having very briefly talked about the background of the Grande Armée, I may FINALLY be able to answer your questions (I believe it is necessary to have at least some basic knowledge of the Organisation of the Grande Armée before being able to answer your question properly, if you want more information on the exact Organisation of the Grande Armée, I will be happy to oblige)

First of all, each country, whether it be ruled by a member of the Bonaparte family or ally of the French Empire, had its own military force which would add up the Napoleon's Grande Armée during his campaigns. I'll detail this further below in my example. In the Kingdoms ruled by Bonaparte family members however, a portion of the French Army would be in garison. For instance, Jospesh (who become King of Spain in 1808) had under his command the General Dupont de l'Etang, who was at the head of about 20,000 soldiers, mostly French. Without going to far into details, for there are many, Dupont's divisions were garisoned near Madrid before surrendering to the Spanish dissidents the same year Jospesh Bonaparte was crowned King of Spain.

Secondly, and more importantly, the Grande Armée recruited men from each Kingdom, country or province under the control of the French Empire. Thus spawned many legendary Legions, such as the Portuguese Legion (roughly 10,000 men), the Polish Legion, German Legion and so on. Paid by the French Empire, they would have a specific status, while still possessing the same rights any other regiment of the Grande Armée had (save perhaps the Guard regiments). Napoleon based the Empire on the basis of equal rights, he would have them enforced in the regiments of the Grande Armée, and each soldier who would fight for the French Empire, wherever he may come from.

As for Vienna and Berlin, being that Austria and Prussia were independent and in total opposition to the French ideals (aka almost consistently at war with the French Empire), Napoleon did not have garison forces in those country's capitals. However, France did station his forces in Berlin following Napoleon's victorious "Campagne de Prusse" in late 1806, although he did it mostly to force the King of Prussia into complying to sign the treaty of Berlin, leading to Continental Blocade (in an effort to diminish Great Britain's power) He withdrawed his forces once he obtained what he deemed necessary.

Finally, I would like to give an example of the way the Grande Armée functionned during one of Napoleon's campaigns. Let's take the Campaign of Russia in 1812.

Napoleon marched towrds Russia with an army of roughly half a million soldiers. I'm not going to talk about the actual battles that occured during this campaign, but more about the way this Grande Armée was organised, more specifically, the "foreign legions".

Napoleon, during this campaign, brought on the top of his elite Imperial Guard, several Corps d'Armée of the French Empire, under the command of his best Maréchaux (Davout, Ney, Berthier for instance) In these Corps d'Armée, we would find the Dutch legion, Polish legion, Swiss Legion and many more. Once again, these legions were recruited and served directly under the command of the French Empire. For instance, the Italian Legion served in the "4eme Corps d'Armée", under the command of Prince Eugène de Beauharnais, son of the Empress Joséphine de Beauharnais (thus not the son of the Emperor by blood)

Additional Corps d'Armées were entirely formed of soldiers from Allied nations, mostly from the Confédération du Rhin, the Army of the Duchy of Warsaw, and even Armies from Austria and Prussia.