No

Unless the prestige class is from Book of Exalted Deeds, Complete Warrior, or Complete Arcane, which specify that those prestige classes do.

The blurbs written in Complete Warrior and Complete Arcane are written as if they were establishing a global rule, but this contradicts the rules for prestige classes in the Dungeon Master's Guide, which mentions no such rule and only states that requirements are necessary to take one’s first level of a prestige class. After the first level has been taken, according to the Dungeon Master’s Guide, all bets are off. You can lose the requirement, and you not only keep the class features, you also retain the right to continue taking levels.

In fact, the Dungeon Master’s Guide includes a prestige class which breaks if this rule is applied: the dragon disciple. The dragon disciple’s requirements include being a non-dragon race, but the capstone of the class applies the half-dragon template, which includes changing the character into a dragon—and thus making it impossible to retain his or her qualification for the class. Under the real rules, this is fine, but under Complete Warrior/Complete Arcane mistakenly applied globally, we end up with Schrödinger’s dragon: gaining the class feature causes the character to lose dragon disciple class features, including the one that broke the requirements, so he or she qualifies again, and then regains the class features and thus breaks them again!

But this is not a problem, because the system includes rules for handling such contradictions.

The errata rules state that when two books contradict each other, the “primary source” wins; Complete Warrior and Complete Arcane are the primary sources for the prestige classes in that book, but the primary rules for the very concept of prestige classes in general is the DMG. So those books can establish an exception for themselves, but not globally. To do that would require a mention in the DMG errata file, which despite reprintings since Complete Warrior and Complete Arcane, has not happened. The DMG was not the only book to print prestige classes that play badly (as in, don’t make sense) with this rule, either.

In contrast, on this subject, Book of Exalted Deeds is written “correctly.” It explicitly supplies an extra rule for its prestige classes (page 49), without trying to assert that such a rule applies to other prestige classes in other books.

In general, gaining prerequisites temporarily is a fairly high-optimization trick; it's probably not welcome at a lot of tables. This is typically more a matter of gentlemen’s agreement than house rule, though.

The reason not to establish this as a rule is because, if, in combat or something, someone is drained or cursed and loses prerequisites, it is not appropriate for every prestige class to have a pseudo “falling” mechanic. The Book of Exalted Deeds prestige classes can explicitly fall and that’s appropriate¹ to that book, but it’s not appropriate for all prestige classes everywhere. It’s better to just say that you don't want people building characters around this trick.

Finally, do consider that the majority of prestige classes have requirements more difficult and painful to meet than they deserve, especially warrior classes which tend to be for the weakest base classes. Thus, if someone is interested in a trick like this, the correct response may easily be: “I don’t want shenanigans like that in this game, but how about we just waive that prereq entirely?” Then everyone’s happy.

1. _{At least, “appropriate” within the context of the official game. I’d personally maintain that falling mechanics are atrociously bad ideas, and should have been implemented far differently. I would have instead written every class that can “fall” as having specific “fallen” replacement features, so that a paladin who falls automatically becomes a blackguard and so on. But that’s all tangential and only my personal opinion.}

Yes, they stack like the rule you cite says, if the specific class says they do. This is an example of exception-based design - the general rule is "only the one class counts" but then the specific rules in the other classes can say "but I do."

Not all prestige classes are alike so you should carefully read their Spellcasting benefit. In this case, Arcane Trickster says "I stack."

But classes that do not say that phrase, like Wizard and Sorcerer, do not stack. In your second example, Loremaster's spellcasting is worded like Arcane Trickster (as if she had gained a level in a spellcasting class she belonged to...), and you would need to decide whether you want it to stack onto your 5th level wizard spellcasting or your completely separate 3rd level sorcerer spellcasting. Assuming the former, you'd cast your spells from wizard slots at level 8 and your ones from your sorcerer slots at level 3.

It's kinda obvious why this is, the core idea behind prestige class design is "continue what you were doing, down a slightly weirder path" and therefore completely penalizing a spellcaster by e.g. halving their spellcaster abilities would be undesirable (even at that, many don't add a spellcasting level at every level and have one or two "penalty" levels.