No
Those "dead" levels are there for a reason. It's supposed to balance the classes. If you want to take the levels beyond that "dead" level, you need to take the dead level first.
You are free to take levels in another class as described in the multi-class rules, if you don't want to advance in that class anymore. But there are no shortcuts to later levels.
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.
Best Answer
The following classes offer full spell-casting progression for either arcane or divine spellcasters.
Arcane Trickster (10 levels) - Requires 2d6 of sneak attack and ability to cast a 2nd level arcane spell. Increases sneak attack and applies it to spellcasting.
Loremaster (10 levels)
Mystic Theurge (10 levels) - Requires 2nd level arcane and divine spells. Increases spellcasting for both an arcane and divine class.
Diabolist (10 levels) - Imp companion
Dissident of Dawn (10 levels) - Requires ability to cast daylight as a divine spell. According to the current FAQ this can be satisfied by a spell-like ability.
Envoy of Balance (10 levels)
Genie Binder (5 levels)
Harrower (10 levels)
Hell Knight Signifer (10 levels)
Soul Warden (10 levels) - Grants channel energy
Spherewalker (5 levels)
Veiled Illusionist (10 levels) - Adds illusion spells from sorcerer/wizard spell list
The following classes offer almost full spell-casting progression for either arcane or divine spellcasters. They give up no more than one level of spell casting progression.
Agent of the Grave (5 levels) - Special abilities for summoned undead
Arcane Savant (7 levels) - Adds spells from any spell list
Demonaic (10 levels) - Demon lord boons
Evangelist (10 levels) - Gains all class features for 9 levels of her aligned class
Arcane
The following classes offer full spell-casting progression for arcane spellcasters.
Bloatmage (10 levels) - Cast spells without using spell slots
Collegiate Arcanist (10 levels) - Requires ability to prepared 3rd-level arcane spells. Adds druid and Good domain to spell list.
Cyphermage (10 levels) - Special scroll abilities
Mage of the Third Eye (10 levels) - Requires arcane school class feature
Natural Alchemist (10 levels) - Grants alchemist discoveries and bombs
The following classes offer almost full spell-casting progression for arcane spellcasters. They give up no more than one level of spell casting progression.
Eldritch Knight (10 levels) - Requires martial weapon proficiency. Gives full BAB progression.
Daivrat (10 levels) - Get one spell from any spell list each day
Technomancer (10 levels)
Winter Witch (10 levels) - Witch only
Divine
The following classes offer full spell-casting progression for divine spellcasters.
Divine Scion (10 levels)
Exalted (10 levels) - Gives Deific Obedience boons, domain spell-like abilities, and a mini miracle.
Green Faith Acolyte (10 levels) - Grants animal companion
Inheritor's Crusader (3 levels) - Only Cleric or Paladin. Full BAB
The following classes offer almost full spell-casting progression for divine spellcasters. They give up no more than one level of spell casting progression.
Balanced Scale of Abadar (6 levels)