Explaining the SSPX controversy
Since people don't know much about the subject but are very interested, let me explain the controversy in a neutral way.
1. Why did the Pope stop everything he was doing to judge the SSPX and declare them excommunicated, when there are so many people committing sinful acts and the Pope doesn't do anything?
He didn't. In Roman Catholicism, you can be excommunicated by the Church after a formal judgment, or you can be automatically excommunicated by the action you perform, such as striking the Pope, procuring an abortion, ordaining a woman to the priesthood, or consecrating a priest as a bishop without a papal mandate.
When they performed that action, they were automatically excommunicated; the Vatican later only issued a report explaining what had happened, rather than issuing a new excommunication.
2. How can the SSPX believe the Pope is infallible, but not accept this decision from him?
As I explained, there was no "decision" this week (though there were in the past); the Vatican merely explained a fact that had already occurred.
Furthermore, the SSPX does believe in the dogma of Papal Infallibility, but that doesn't mean the Pope is infallible, but merely that the Pope can protect the faith in some circumstances infallibly, by making a pronouncement to be viewed as definitive, but there are rare pronouncements.
3. How can the SSPX believe that Vatican II is a valid council if they oppose it, considering that valid councils are infallible?
A council's infallibility applies to its theological proclamations, not necessarily to every single thing said within it. A council is also a way for the Church to implement changes in governance with the participation of all bishops worldwide.
Let me give you an example using a council we all agree on: the Council of Nicaea. It includes a proclamation of what the faith is (the Nicene Creed), but it also contains a bunch of governing rules about how the Church ought to operate. For instance, the first canon forbids men from becoming priests if they have self-castrated. Such governing rules can be considered changed later under new circumstances, whereas the Nicene Creed is infallible.
The SSPX views the doctrinal affirmations of Vatican II as infallible, as they believe those parts merely repeated earlier definitions. However, they believe the governing rules and disciplinary changes defined in the council are flawed and need to be changed. The most obvious example is the change in the liturgy, but there are others, such as how the Church promotes the human concept of freedom of religion.
4. Even if the Pope is not speaking infallibly, since the SSPX views him as a valid Pope with ordinary authority, why don't they obey him?
The SSPX argues that due to the current crisis of faith in the Church, they are allowed to bypass canonical rulings. They cite Canon 1752 of the 1983 Code of Canon Law, which states that "the salvation of souls is the supreme law," and they maintain that this is the ultimate framework in which all other laws must be observed.
5. Why didn't the Pope just declare this priestly fraternity dissolved?
He did, in 1975 (the bishop who gave formal approval was overruled, and the Pope agreed), but the fraternity continued to operate unlawfully. They maintain that the crisis of faith in the Church makes their continued existence necessary and justified.
If you want to know more about the details, this video of a Ready to Haverst (an independent baptist) explain it in a neutral way: