Good job, Trevize!
The Second Seldon Crisis did come early, but it was not Hardin's fault.In The Mayors, Salvor Hardin worries that the internal crisis - a group demanding action and hostile to him personally taking control of the Council - and the external crisis - the most powerful of the Four Kingdoms attacking - were supposed to come close to one another, but the external crisis is very likely to come months earlier. The thing is, the main flaw with psychohistory is that it accounts for human masses, and by nature cannot account for freak incidents - the larger the population, the less important that becomes, but at that point in time, the effective 'universe' to the Foundation was Terminus and the Four Kingdoms, and a chance discovery of an old Imperial cruiser had been made: the foreign incident caused by that chance discovery threw the calculations off, not enough to make the Seldon Plan fail, but enough that there was a risk of it failing without sufficiently capable individual leadership on the Foundation's side. Salvor Hardin's slight foresight might even have helped him put things back as they should be!
The Second Foundation's main Plan-disturbing problem in the years after the Kalgan War is a feature of the Plan.We are told - and shown - that one of the points of the First Foundation was to maintain a vibrant technical scientific tradition that it would build upon. The problem is that while trends in research might be predicatable from aggreggate human behaviour - the early Foundation having a heavy bias towards miniaturisation and efficiency, for example - specific discoveries are not, and sometimes science throws up something that radically changes things (indeed, psychohistory itself is proof of that). Hari Seldon could not predict if the First Foundation around the middle-mark would develop, say, a technology that would drastically cut interplanetary and interstellar travel-times, so the Second Foundation has to step in and ensure that any such discoveries does not throw the Plan off/incorporate the discovery into the Plan.
The Seldon Plan failed, but it doesn't matter.The conquests of the Mule brought order spread modern technology throughout most of the Galaxy. Even after his death, the Galaxy doesn't revert to the chaos and barbarism of the pre-Mule era, and the Foundation is no longer the sole centre of science and technology. The Galaxy will eventually be reunified, but not by any one power. Instead, it will develop into a loose confederation of major powers, of which the Foundation will be only one. Given that Foundation is a retelling of the fall of the Roman Empire IN SPACE, this would parallel the development of Medieval and Renaissance Europe. No one European power could replace the Roman Empire, but that didn't mean an end to stability and culture.
Tazenda was not razed by the Mule's forces.The First Speaker needed to get the Mule to be as emotionally confident as was possible in order to make the fall all the more dramatic. Thus, he played along and claimed that the Second Foundation had merely lessened the devastation caused by the Mule's fleet, when in reality they had prevented it altogether.
R. Daneel Olivaw and his followers are the Danellians of Time PatrolA group of people did invent time travel, and R. Daneel saw it as a threat to his plans. So, he and his followers got their hands on some time-traveling machines, and set up the Time Patrol. They made up some details about themselves and rarely appeared in front of patrol members (and used special effects/some sort of interference with their minds to appear as they are described in Time Patrol). No one but the most highly ranked patrol members were allowed to know the full truth, because they may not be as willing to do their jobs if they knew that the future they were fighting for might potentially involve something like Gaia. Their ideas about what would be best for humanity might have been different from those of R. Daneel. The robots also give the Patrol a lot of autonomy since they have so many other areas to control. The Time Patrol is also tasked with finding out about the past since most pre-Galactic Empire records have been destroyed, and Gaia can only remember back as far as the time it was created. The information gathered by the Time Patrol can be studied by Gaia to add to Gaia's knowledge of history and human nature, which may be useful at some point in the future.
Both of the Mule's origin stories are trueGaians do operate off Gaia (e.g. Sura Novi on Trantor) and potentially a "Gaian" could be born anywhere in the galaxy. The Mule could indeed be a Mutant, but contrary to what everybody believes, his mentallic powers are not his mutation. His real mutation (besides his deformity) is that he is not connected to the rest of Gaia, and for some reason could not be. Gaia might have tried to assimilate him and found itself unable to do so. This would account for his aversion to the planet. As for his powers, those are simply the abilities that all Gaians possess.
The Seldon Plan and Galaxia are complementary, not competitiveSeldon was in a desperate rush to complete the basic Plan before he died of old age. In prior conversation with the Emperor, he had expressed sympathy for the idea of democracy, but conceded that there was no way to make it work on a world as densely-populated as Trantor, much less throughout the entire Empire. So he focused his efforts towards a goal for which he possessed a working mathematical model, a second Galactic Empire. It was not in fact the best possible future for humanity, just the best he had time to run the numbers for. The question of what direction the Plan would take after the establishment of the new Empire he left largely in the hands of the Second Foundation, who were planning to become a Supernatural Elite ruling from behind the scenes, literally changing people's minds as needed. But Daneel is the chessmaster playing the game across the galaxy. Gaia is a success, but has limits. Most notably, it cannot just assimilate anyone. Children possibly. But adults like Janov Pelorat cannot join the superorganism even if they really want to. Hence the Seldon Plan. A second Empire unifying the galaxy is a crucial stepping stone to Galaxia. Just as Daneel taught the Gaians the secrets of mentalics centuries earlier in order to create the group mind, it would be much easier to do the same to the entire galactic population if it were already unified under one government. Thus, at the end of Foundation's Edge, Gaia wipes Mayor Branno of her ambitions of making a premature attempt to build the new Empire centuries early. Stor Gendibel is likewise mind-wiped of the events at Gaia, and returned to the task of advancing the Seldon Plan, and not coincidentally working to build a greater mental fusion than they have now. Gaia has set the Plan back on track. Had Seldon lived long enough to run his calculations far beyond the establishment of the Second Empire he would have probably discovered Gaia/Galaxia. The Second Foundation almost found it when Stor Gendibel identified the "anti-Mules" that were seemingly doing even more to keep the Plan on track than the Second Foundation. The Second Galactic Empire will be born, and will run for hundreds, perhaps thousands, of years while the Gaians gradually introduce mentalics to the population and construct the galaxy-spanning superorganism.