physical chemistry - Predominance of III oxidation state for lanthanides

For most lanthanide metals, the stable oxidation state is III [*]. The general electronic structure is $$\ce{[Xe] 4f^{0}^{-14} 5s^2 5p^6 5d^{0}^{-1} 6s^2}\ \ [**].$$

Elements that have the d-electron are La, Ce, Gd, and Lu. Furthermore, the f-subshell is considered relatively stable in states $$f^0, f^7, \text{and} f^{14}.$$

We can conclude that La, Gd, and also Lu easily form $\ce{E^3+}$ ions. Yet, as would be predicted by this easy approach, we would also see $$\ce{Sm+, Tm+ (f^7 \ and \ f^14 ), \\ Pr^5+, Dy^5+ (f^0 \ and \ f^7)}.$$

This is definitely not the case. But the why? alludes me. According to Radiochemistry,

The 4f binding energy is so great that the remaining 4f-electrons are regarded as "core-like". [Italic added]

Could someone explain this further? [***]

Radiochemistry then goes on to assert,

Interelectronic repulsion in related not just to electron pairing, but also to [the] angular momentum of the electrons. [---]

e.g., in $\ce{Pr^2+ (4f^3)->Pr^3+ (4f^2)}$ ionisation removes repulsion between [electrons] of like rotation [---]

$\ce{Pm^2+ (4f^4)->Pm^3+ (4f^3)}$ ionisation removes repulsion between [electrons] of unlike rotation [---] [emphasis added]

What exactly is meant by "electrons of like/unlike rotation"? Plus it mentions some "quarter / three-quarter" rule which I possibly cannot rationalise for a total of 14 electrons.

Any help would be greatly appreciated.

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[*] Ce, Pr, and Tb also have the oxidation state IV. Eu and Tm have the additional state II.

[**] Ordering changes with the number of electrons.

[***] I am, of course, aware that

• each electron would be harder to remove as Culonic attraction increases with positive ionic charge,


• lathanide contraction is a part of the explanation.

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(Somewhat related: Plutonium having more oxidation states than samarium?)

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EDIT

Every thanks to Brian for providing awesome title suggestions. These are the four new subquestions:

Just an FYI for future chainposters, there's a sixty-second waiting period after asking two consecutive questions.