How I do it/think it should work...
1. Yes. Each Class should have their spell lists listed - preferably with a brief descriptive, no mechanics - in their Class pages/section. It gives a potential player immediate access to "look at the kinds of things you could/will be able do." It lends immediate flavor to the class.
2. Yes. There still needs to be a "Spells/Magic" section to a player's manual, to give all of the details, mechanics, expanded description, and/or expanded uses (a 5th level caster can then do/get X from this spell, a 9th level caster can then do/get Y, etc...)
3. No. There is no reason every spell list needs be entirely original....Clerics - in my game - get Cure Wounds (I) as a 1st tier spell. Druids don't get Cure Wounds as a 2nd tier spell, which they don't have access to until 4th level, (I think, off the top of my head). Bards have it on their spell lists. Necromancers. I even let Abjurists have some minor healing.
4. No. The same spell appearing on different spell lists need not be of the same spell level or accessed at the same levels of experience for all classes.
5. I am also a negatory that all casters be created equal. All spell lists, even for the same "type" of magic (divine, arcane, nature/elemental,... psychic, occult/eldritch...whatever distinctions you use/like or however you want to break them up) don't have to be the same length. The Wizard/Mage is supposed to be THE caster class. The Best there is at using magic. That's their niche. That's their role in the game and party. They cast the spells. That's what they -primarily- bring to the table. That class should have a spell list that far exceeds any other class' (maybe the Druid comes close). Just for sheer number of options. Another "arcane magic"-using class, a Swordmage/Bladecaster type - accessing the same kinds of spells, using the same kinds of preparation/study (or however you narrate your arcane casters), but - is a niche class. They cast spells with their sword/weapon swings, primarily for use IN close combat situations. Their spell list shouldn't be nearly the panoply of utilitatrian/defensive/offensive options a straight up Wizard has access. Some overlap (Magic Missle, Shield, etc...), sure. Some unique to the Swordmage "magic up my weapon" or "control my section of the battlefield" spells, sure. But not as many as a Wizard. Does the Swordmage need "Charm Person" on their list? Not really, no. Does an Enchanter/Illusionist specialist? Does a Conjurer?