As an author of one of these Napoleon-wins-TLs, I might be slightly biased towards the direction that my TL took over the others, but hopefully I can help anyway:
It's not uncommon to see the French Empire continue to dominate Europe and even rise to the status of superpower later, even continuing until the 21st century.
Why shouldn't it be common? The second nation to industrialise after Britain was Belgium, and any Napoleonic victory will see Belgium be part of France. Most likely France also has the resources of the Rhineland too.
There's also the fact that lots of monarchs were tipped out and replaced by other Bonapartes, so at least until the first post-Napoleon war France is certain to exert a substantial amount of control over many of its neighbours. OTL Germany, Italy and France were 3 of the 6 strongest powers in Europe during WWI, and combining them into effectively one state would just amplify this fact. I know that 1914 isn't the same as 1814, but it isn't too drastically different either.
In some ways, the imagined history of Europe and France seems often almost stagnant compared what we actually experienced.
Between 1815 and 1860 or so, European history was pretty stagnant, with the exception of a couple of outbreaks of revolutions, most notably 1848. If Napoleonic France is still strongarming its neighbours to have loyal governments, then the revolutions most likely become either very localised affairs (as my TL's German Revolutions c. 1840) or don't rise at all. Revolutions only tend to occur in periods when 1) the government is seen as decadent/corrupt/the leader is an idiot
and 2) a viable alternative is seen, whether it be another ruler or a totally different government type. The former is certainly possible (at least in idiot leader form) in any TL, so I won't address that further. However the latter is harder to achieve. For example, take a nation like Prussia:
- Replace the king with someone that doesn't like Napoleon? France will probably just invade and stick the old guy back in.
- Replace the monarchy with a republic? In a Napoleon TL, there isn't a proven history of republics being successful in big countries, with only the USA as an example. If the USA is weaker in the ATL, that isn't seen as a good idea.
- Replace with a new ideology? Possible, but new ideologies don't spawn all that frequently, and if it gets discredited then that option's gone too.
Firstly, how likely is that the French Empire would avoid upheavels in any way comparable to those that its OTL counterpart did?
Honestly, a French Empire in the 1820s will be more stable simply due to Napoleon himself. If he wins the wars for good, especially with a PoD past 1809, he is seen as basically a living god by the people of France. No-one is going to question him any more. Note that between the time of the Munich Pact and late 1941, very few people questioned Hitler. This was simply because he was successful. Even when the decisions he made were pretty stupid (notably the Blitz), there were few objections even considered. Why? Because he was winning, and no-one wants to go against someone who is regularly seen as a winner. Napoleon post 1812 will be this but x10.
Simply put, how likely is it that the French Empire would still exist in 2017, instead of some alternative Third French Republic?
2017 is too far forward to comment with any real basis of meaning, because each TL will have its own history by that point. By a more reasonable date, say 1850, Napoleonic France's chances are pretty good if its leaders (not just the Bonapartes, but generals, industrialists, imperial advisors &c) are at least semi-competent. The state was basically a superpower by 1812, and there's no inherent reason why that can't continue to be the case for at least a little while. If they do get plagued with a moronic leader or a stack of bad luck (and any good TL should see the majority of countries hit with periods of bad luck, although they also need good luck from time to time too), then it is equally reasonable that they fall.
Secondly, how likely it is that France would be still a leading power in Europe or the world by the 21st century? OTL history did show fluctuations in nations' fortunes during last centuries. Would France's territorial extent still be similar to that it had in 1812, or is it possible it could lose some of it?
Depends on the definition of "borders of 1812", but the short answer is that they will probably lose some of it. If we're talking about the system of puppet states (or as it becomes known in my TL, "the Alliance"), then that will eventually fall for one reason or another, whether it be war, monarchs hating each other, revolutions or whatever. No alliance ever lasts for two centuries uninterrupted.
If we ignore Illyria and Rome for a moment, France could probably hold on to the rest of its lands if there doesn't develop some huge revanchist movement (my TL is notable for its revanchism!). The Rhine border, the Netherlands and Catalonia were all close enough to Paris to be effectively governed and the people were culturally similar enough to integrate after a generation or two.
I have noticed that there often seems to exists this idea of relatively stable European/French political system following Napoleon's victory.
Haha. I wouldn't call my work "stable" (at least for the 19th century anyway). Just curious, have you read my TL?
Beyond what you have asked for, I feel the following two points need to be considered in relation to how much the success of a Napoleonic system would be:
- PoD: A war ending in 1805/6 will mean a more localised France than one that ends in 1812/3. Note that a PoD in 1805 but with similar Napoleonic victories until 1812 will mean that you get a more 1812-PoD world than if everyone just stopped fighting after Austerlitz or Tilsit.
- The outcome of whatever war happens in the 1830s or so, or even whether one happens at all. If Napoleon dies and his son beats Europe, France is assured success for a good half-century or more. If he loses, France might retain some influence over Germany and/or Italy, but to nowhere near the extent that France did in OTL 1810.
- BNC