A. Let $p$ be a prime and $G=C_p$ the cyclic group of order $p$. If $p=2$, the answer to your question is yes and if $p$ is odd, then it is no.
First let me rephrase the question. Fix $f\colon X\to Y$ and define a full subcategory $\mathcal{C}_f \subset \mathrm{Sp}_G$ of $G$-spectra as follows: If $f_*\colon [A,X]\to [A,Y]$ is an isomorphism, then $A\in \mathcal{C}_f$. Note that $\mathcal{C}_f$ is a localizing subcategory.
Let $\mathrm{Lin}\subset \mathrm{Pic}(\mathrm{Sp}_G)$ be the linear representation spheres. We can now rephrase your question in either of the following forms:
- Does the following implication hold for all $f$: $\mathrm{Lin}\subset \mathcal{C}_f\implies {C_p}_+\in \mathcal{C}_f$?
- Is $\mathrm{Sp}_G$ the smallest localizing subcategory containing $\mathrm{Lin}$?
When $p=2$ and $\sigma$ is the real sign representation, then the cofiber sequence: $$ {C_2}_+ \to S^0\to S^{\sigma} $$ shows 1. holds.
Suppose that $p$ is odd and 2. holds. Let $g\colon Z\to {C_p}_+$ be a $\mathrm{Lin}$-cellularization of ${C_p}_+$, i.e., $g$ is a $G$-equivalence and $Z$ is constructed by iteratively gluing 'cells' from $\mathrm{Lin}$ (just as in a CW-approximation). Taking non-equivariant rational homology we see that the permutation $C_p$-representation $\mathbb{Q}[C_p]$ is in the localizing subcategory generated by the trivial representation (since $p$ is odd, the homology of the representation spheres is always a trivial module). We are unable to build a non-trivial action from trivial ones, so this is a contradiction.
I do not have a good answer for the remaining questions. For general groups there are many indices in the $RO(G)$-grading. Even if you pass to the smaller $JO(G)$-grading, I would think it would be difficult to convey the data concisely. For cyclic $p$-groups (and perhaps more), one can probably get at the $RO(G)$-graded groups for an Eilenberg-MacLane functor. Ferland-Lewis and Lewis have relevant calculations. I imagine there are more calculations when $G=C_2$.