As for 2) it is not clear to me if you mean homotopy limits or colimits. If you mean homotopy limits I think you are in trouble; products are particular examples and the kind of localisation (of homology) that you need does not commute with products. For homotopy colimits you should be in better shape. There is however an initial problem: If you do not assume that the particular group completions you choose have any functorial properties it is not clear that a diagram over a category will give you a diagram when you group complete. This can be solved by either assuming that in your particular situation you have enough functoriality to get that (which seems to be the case for for instance May's setup) or accepting "homotopy everything" commutative diagrams which you should get by the obstruction theory above. If this problem is somehow solved you should be able to conclude by the Bousfield-Kan spectral sequence $\injlim^\ast H_*(X_i,\mathbb Z)\implies H_*(\mathrm{hocolim}X_i,\mathbb Z)$. We have that localisation is exact and commutes with the higher derived colimits so that we get upon localisation a spectral sequence that maps to the Bousfield-Kan spectral sequence for $\{Y_i\}$ and is an isomorphism on the $E_2$-term and hence is so also at the convergent.

Addendum: As for 2) it seems to me that this question for homotopy limits can only be solved under supplementary conditions. The reason is that under some conditions we have the Bousfield-Kan spectral sequence (see Bousfield, Kan: Homotopy limits, completions and localizations, SLN 304) which shows that $\varprojlim^s(\pi_s X_i)$ for all $s$ will in general contribute to $\pi_0$ of the homotopy limit. As the higher homotopy groups can change rather drastically on group completion it seems difficult to say anything in general (the restriction to cosimplicial spaces which the OP makes in comments doesn't help as all homotopy limits can be given as homotopy limits over $\Delta$. Incidentally, for homotopy colimits you should be in better shape. There is however an initial problem (which also exists in the homotopy limit case): If you do not assume that the particular group completions you choose have any functorial properties it is not clear that a diagram over a category will give you a diagram when you group complete. This can be solved by either assuming that in your particular situation you have enough functoriality to get that (which seems to be the case for for instance May's setup) or accepting "homotopy everything" commutative diagrams which you should get by the obstruction theory above. If this problem is somehow solved you should be able to conclude by the Bousfield-Kan spectral sequence $\injlim^\ast H_*(X_i,\mathbb Z)\implies H_*(\mathrm{hocolim}X_i,\mathbb Z)$. We have that localisation is exact and commutes with the higher derived colimits so that we get upon localisation a spectral sequence that maps to the Bousfield-Kan spectral sequence for $\{Y_i\}$ and is an isomorphism on the $E_2$-term and hence is so also at the convergent.

Starting with 1) a first note is that your conditions does not have to involve an arbitrary ring $R$. It is enough to have $R=\mathbb Z$ and one should interpret the localisation in the way (for instance) Adams does: $H_\ast(X,\mathbb Z)=\bigoplus_\alpha H_\ast(X_\alpha,\mathbb Z)$, where $\alpha$ runs over $\pi_0(X)$, and a $\beta$ maps $H_\ast(X_\alpha,\mathbb Z)$ to $H_\ast(X_{\alpha\beta},\mathbb Z)$. Then your group completion condition is that the natural map $\mathbb Z[\pi_0(Y)]\bigotimes_{\mathbb Z[\pi_0(X)]} H_\ast(X,\mathbb Z)\rightarrow H_\ast(Y,\mathbb Z)$ should be an isomorphism. This then implies the same for any coefficient group (and when the coefficient group is a ring $R$ you get your condition).

Starting with 1) a first note is that your conditions does not have to involve an arbitrary ring $R$. It is enough to have $R=\mathbb Z$ and one should interpret the localisation in the way (for instance) Adams does: $H_\ast(X,\mathbb Z)=\bigoplus_\alpha H_\ast(X_\alpha,\mathbb Z)$, where $\alpha$ runs over $\pi_0(X)$, and a $\beta$ maps $H_\ast(X_\alpha,\mathbb Z)$ to $H_\ast(X_{\alpha\beta},\mathbb Z)$. Then your group completion condition is that the natural map $\mathbb Z[\pi_0(Y)]\bigotimes_{\mathbb Z[\pi_0(X)]} H_\ast(X,\mathbb Z)\rightarrow H_\ast(Y,\mathbb Z)$ should be an isomorphism. This then implies the same for any coefficient group (and when the coefficient group is a ring $R$ you get your condition). (Note that for this formula to even make sense we need at least associativity for the action of $\pi_0(X)$ on the homology. This is implied by the associativity of the Pontryagin product of $H_*(X,\mathbb Z)$ which in turn is implied by the homotopy associativity of the H-space structure.)