I can answer your first question in some special cases. 

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:
A $G$-spectrum $A$ is in $\mathcal{C}_f$ precisely when $$f_*\colon [\Sigma^n A,X]\to [\Sigma^n A,Y]$$ is an isomorphism for all $n\in \mathbb{Z}$. Note that $\mathcal{C}_f$ is a localizing subcategory (i.e., it is a thick subcategory and closed under arbitrary coproducts). 

Let $\mathrm{Lin}\subset \mathrm{Pic}(\mathrm{Sp}_G)$ be the linear representation spheres (i.e, the $G$-spectra of the form $S^{V-W}\simeq S^V\wedge DS^W$ where $V$ and $W$ are real $G$-representations). We can now rephrase your question in either of the following forms: 

 1. Does the following implication hold for all $f$: $\mathrm{Lin}\subset \mathcal{C}_f\implies {C_p}_+\in \mathcal{C}_f$?
 2. Is $\mathrm{Sp}_G$ the smallest localizing subcategory containing $\mathrm{Lin}$?


Indeed by definition, a map $f\colon X\to Y $ is a $G$-equivalence, if it induces an isomorphism $$f_*\colon [\Sigma^* G/H_+, X]\to [\Sigma^* G/H_+, Y],$$ for every (closed) subgroup $H\subset G$. When $G=C_p$ the only subgroups are the trivial subgroup and the whole group $G$. Under the hypothesis of 1. the map $f_*$ is an isomorphism by assumption when $H=G$, so we just need to show $G/e_+={C_p}_+\in \mathcal{C}_f$. This directly relations to condition 2.: The smallest localizing category containing $S^0\in \mathrm{Lin}$ and ${C_p}_+$ is $\mathrm{Sp}_{C_p}$.


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 ${C_p}_+$. In other words, it is a colocalization with respect to $\mathrm{Lin}$; $Z$ is constructed by iteratively gluing 'cells' from $\mathrm{Lin}$ (just as in a CW-approximation) and $g$ induces an isomorphism on $RO(G)$-graded homotopy groups. Since we are assuming 2. $g$ is actually a $G$-equivalence.

Now forgetting down to spectra with a $C_p$-action (i.e., the homotopy theory of $C_p$-diagrams in spectra or the $\infty$-category $\mathrm{Fun}(BC_p, \mathrm{Sp}))$ and taking rational homology we obtain an isomorphism of two graded $\mathbb{Q}$-vector spaces with $C_p$-actions. This shows 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 shifted copy of $\mathbb{Q}$ with a trivial action). We are unable to build a non-trivial action from trivial ones, so this is a contradiction.

If instead of working over the sphere, we work in the category of $KU_G$-modules, then I can say a bit more. Let $G$ be a connected compact Lie group with $\pi_1 G$ torsion-free. Then it is a result of Akhil Mathew, Niko Naumann, and myself that a map $f\colon X\rightarrow Y$ of $KU_G$-modules is an equivalence precisely when it induces isomorphisms: $$ f_*\colon [\Sigma^* S^0, X]\to [\Sigma^* S^0, Y].$$ So here, you don't even need all of the representation spheres, just the trivial representations. This result follows from Thm. 8.3 in [Nilpotence and descent in equivariant stable homotopy theory][1].

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 $G=C_2$.


  [1]: http://arxiv.org/abs/1507.06869 "Nilpotence and descent in equivariant stable homotopy theory"