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.