Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.

Also, I've looked through other similar MO questions, but I didn't find answers to the questions below. Of course, please let me know if I missed anything!

In motivic homotopy theory, the category of motivic spectra comes with a set of "bigraded spheres", $S^{i,j} = \Sigma^i (\mathbb{G}_m)^{\wedge j}$. It is well-known that homotopy classes of maps out of these don't detect equivalences. This gives rise to a notion of "cellular" motivic spectra (as studied by Dugger--Isaksen): these sit as a right localization -- that is, a full subcategory whose inclusion admits a right adjoint -- which is by definition the largest subcategory in which bigraded homotopy groups do detect equivalences.

Back in equivariant homotopy theory, the analogous "obvious Picard elements" are the virtual representation spheres $S^V$, i.e. $V$ is a formal difference of finite-dimensional $G$-representations. These corepresent what are therefore known as "$RO(G)$-graded homotopy groups".

A. If a map $X \to Y$ of $G$-spectra induces an isomorphism $[S^V,X] \xrightarrow{\cong} [S^V,Y]$ for all $V \in RO(G)$, is it necessarily an equivalence?

I would of course be interested to hear partial answers, e.g. if this is only known when $G$ is finite/discrete but is open for $G$ a (compact) Lie group -- similarly if this depends on what $G$-universe I'm working over (though the answer is pretty obviously "no" e.g. for the trivial universe).

Moreover, if the answer to the above is "no", then I have a follow-up question.

A'. What is known about the subcategory of cellular $G$-spectra? Is it known to contain or not contain any particular $G$-spectra of interest?

Another question I have is the following.

B. Are there any particular $G$-spectra of interest with known $RO(G)$-graded homotopy groups?

In my limited understanding, the more common notion of "homotopy groups" in the equivariant world are given by homotopy classes of maps out of the stabilized orbits $\Sigma^n (G/H)_+$ (which altogether assemble into a Mackey functor). By definition, these detect equivalences. (The motivic analog would be mapping out of $\Sigma^\infty X_+$ for all $X$ in the Nisnevich site.)

C. Besides the obvious ones, are there any known relationships between $RO(G)$-graded homotopy groups and the "homotopy" Mackey functor?

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.

Also, I've looked through other similar MO questions, but I didn't find answers to the questions below. Of course, please let me know if I missed anything!

In motivic homotopy theory, the category of motivic spectra comes with a set of "bigraded spheres", $S^{i,j} = \Sigma^i (\mathbb{G}_m)^{\wedge j}$. It is well-known that homotopy classes of maps out of these don't detect equivalences. This gives rise to a notion of "cellular" motivic spectra (as studied by Dugger--Isaksen): these sit as a right localization -- that is, a full subcategory whose inclusion admits a right adjoint -- which is by definition the largest subcategory in which bigraded homotopy groups do detect equivalences.

Back in equivariant homotopy theory, the analogous "tautological Picard elements" are the virtual representation spheres $S^V$, i.e. $V$ is a formal difference of finite-dimensional $G$-representations. These corepresent what are therefore known as "$RO(G)$-graded homotopy groups".

A. If a map $X \to Y$ of $G$-spectra induces an isomorphism $[S^V,X] \xrightarrow{\cong} [S^V,Y]$ for all $V \in RO(G)$, is it necessarily an equivalence?

I would of course be interested to hear partial answers, e.g. if this is only known when $G$ is finite/discrete but is open for $G$ a (compact) Lie group -- similarly if this depends on what $G$-universe I'm working over (though the answer is pretty obviously "no" e.g. for the trivial universe).

Moreover, if the answer to the above is "no", then I have a follow-up question.

A'. What is known about the subcategory of cellular $G$-spectra? Is it known to contain or not contain any particular $G$-spectra of interest?

Another question I have is the following.

B. Are there any particular $G$-spectra of interest with known $RO(G)$-graded homotopy groups?

In my limited understanding, the more common notion of "homotopy groups" in the equivariant world are given by homotopy classes of maps out of the stabilized orbits $\Sigma^n \Sigma^\infty_+ (G/H)$ for $n \in \mathbb{Z}$, which altogether assemble into a Mackey functor. By definition, these detect equivalences. (The motivic analog would be mapping out of $\Sigma^{\infty+n} X_+$ for all $X$ in the Nisnevich site.)

C. Besides the obvious ones, are there any known relationships between $RO(G)$-graded homotopy groups and the "homotopy" Mackey functor?