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Related post: this

This is a related question to (3).
This post shows (3) (c). As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories. But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties. (e.g., this post and this pdf.) Are there any other important propositions of Grothendieck groups?

I've heard that we can show the Kunneth's formula more easily using derived categories. I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)

Related post: this (a) and this(c).

This is a related question to (3).
This post shows (3) (c). As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories. But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties. (e.g., this post and this pdf.) Are there any other important propositions of Grothendieck groups?
And would you give me some references?

I've heard that we can show the Kunneth's formula (of schemes) more easily using derived categories. (see here and for singular cohomology here.) I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)

(1) Are the derived categorical derived functors universal in the classical sense?
Let $\mathscr{A, B}$ be two abelian categories, $f : \mathscr{A} \to \mathscr{B}$ a left exact functor, and assume that $\mathscr{A}$ has enough injective. Then there exists "the" right derived functor $\mathbb{R}^+f: D^+(\mathscr{A}) \to D(\mathscr{B})$ of $f$, and its $i$-th cohomology of an object $X$ in $\mathscr{A}$ (considering as the complex which has $X$ at $0$-th degree) is the classical $R^if(X)$.
So $\{ H^i(\mathbb{R}^+f(X)) \}_i$ is universal, in the sense of Hartshorne's AG, chapter III, i.e., for every $\delta$-functors $\{ g^i \}_i$ from $\mathscr{A}$ to $\mathscr{B}$ (i.e., a collection of additive functors satisfying the following condition: for every short exact sequence $0 \to X \to Y \to Z \to 0$ in $\mathscr{A}$, there exists the "connection map" $g^iZ \to g^{i+1}X$, making the long sequence exact.), if we have a natural transformation $f \to g^0$, there exists a unique natural transformations $R^if \to g^i$.
Now is there a derived categorical interruption of this phenomena? I.e., for such $\{g^i\}$ and $f \to g^0$, does there exist a $\delta$-functor $g: D^+(\mathscr{A}) \to D(\mathscr{B})$ (such that for every $X \in \mathscr{A}$, $H^i g(X) = g^i(X)$) and $\mathscr{Q}f \to g \mathscr{Q}$? ($\mathscr{Q}$ is the localizing functor.) If so, then by the universality (in derived categorical sense), we have $\mathbb{R}^+f \to g$.
(I read this post, but it seems to be a bit different from my question.)

(2) Can we show the "Grothendieck's Tohoku" easily using derived category?
This theorem says that, in particular, for a $\delta$-functor $\{g^i\}$, if $g^i(I) = 0$ for every $i \gt 0$ and every injective object $I$, then this is universal, i.e., $g^i \cong R^i g^0$.
If (1) is true, then I think that this theorem can be translated into the following form:

(3) How can we use derived categorical derived functors in order to show propositions around spectral sequences?
I'm studying derived categories by Hartshorne's "Residues and duality". In this text, the author says "What used to be a spectral sequence is now simply a composition of functors. And of course one can recover the old spectral sequence..." (see the remarks after the Proposition 5.4.)
I think the author means that we can show "all" propositions which are used to be shown by the spectral sequence argument using derived categories.
For example:

(4)Grothendieck group of a derived category
This is a related question to (3).
This post shows (3) (c). As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories. But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties. (e.g., this post and this pdf.) Are there any other important propositions of Grothendieck groups?

Lastly,
(5)Easier proof of classical homological propositions
I've heard that we can show the Kunneth's formula more easily using derived categories. I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)

Thank you very much!

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(1) Are the derived categorical derived functors universal in the classical sense?

Let $\mathscr{A, B}$ be two abelian categories, $f : \mathscr{A} \to \mathscr{B}$ a left exact functor, and assume that $\mathscr{A}$ has enough injective. Then there exists "the" right derived functor $\mathbb{R}^+f: D^+(\mathscr{A}) \to D(\mathscr{B})$ of $f$, and its $i$-th cohomology of an object $X$ in $\mathscr{A}$ (considering as the complex which has $X$ at $0$-th degree) is the classical $R^if(X)$.
So $\{ H^i(\mathbb{R}^+f(X)) \}_i$ is universal, in the sense of Hartshorne's AG, chapter III, i.e., for every $\delta$-functors $\{ g^i \}_i$ from $\mathscr{A}$ to $\mathscr{B}$ (i.e., a collection of additive functors satisfying the following condition: for every short exact sequence $0 \to X \to Y \to Z \to 0$ in $\mathscr{A}$, there exists the "connection map" $g^iZ \to g^{i+1}X$, making the long sequence exact.), if we have a natural transformation $f \to g^0$, there exists a unique natural transformations $R^if \to g^i$.
Now is there a derived categorical interruption of this phenomena? I.e., for such $\{g^i\}$ and $f \to g^0$, does there exist a $\delta$-functor $g: D^+(\mathscr{A}) \to D(\mathscr{B})$ (such that for every $X \in \mathscr{A}$, $H^i g(X) = g^i(X)$) and $\mathscr{Q}f \to g \mathscr{Q}$? ($\mathscr{Q}$ is the localizing functor.) If so, then by the universality (in derived categorical sense), we have $\mathbb{R}^+f \to g$.
(I read this post, but it seems to be a bit different from my question.)

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(2) Can we show the "Grothendieck's Tohoku" easily using derived category?

This is related to (1).
This theorem says that, in particular, for a $\delta$-functor $\{g^i\}$, if $g^i(I) = 0$ for every $i \gt 0$ and every injective object $I$, then this is universal, i.e., $g^i \cong R^i g^0$.
If (1) is true, then I think that this theorem can be translated into the following form:

-

(3) How can we use derived categorical derived functors in order to show propositions around spectral sequences?

I'm studying derived categories by Hartshorne's "Residues and duality". In this text, the author says "What used to be a spectral sequence is now simply a composition of functors. And of course one can recover the old spectral sequence..." (see the remarks after the Proposition 5.4.)
I think the author means that we can show "all" propositions which are used to be shown by the spectral sequence argument using derived categories.
For example:

-

(4) Grothendieck group of a derived category

This is a related question to (3).
This post shows (3) (c). As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories. But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties. (e.g., this post and this pdf.) Are there any other important propositions of Grothendieck groups?

-

Lastly, (5) Easier proof of classical homological propositions

I've heard that we can show the Kunneth's formula more easily using derived categories. I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)

Any help will be much appreciated!!

Lastly,
(4)Easier proof of classical homological propositions
I've heard that we can show the Kunneth's formula more easily using derived categories. I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)

(4)Grothendieck group of a derived category
This is a related question to (3).
This post shows (3) (c). As we can see from this post, it seems to be very important to understand Grothendieck groups of derived categories. But I don't know any references of Grothendieck groups of derived categories, especially of algebraic geometrical objects.
I've found some fundamental properties. (e.g., this post and this pdf.) Are there any other important propositions of Grothendieck groups?

Lastly,
(5)Easier proof of classical homological propositions
I've heard that we can show the Kunneth's formula more easily using derived categories. I want to know such propositions. Would you give me references?
I also want references of derived categorical proof of cohomology and base change theorems(see III.12 of Hartshorne's AG or here. The later statement is too abstract for me. I prefer Hartshorne-like concrete statement.)