Here is a solution inspired by Losif Pinelis, but addresses the existence of non-singular matrices (see comment to last answer).

First consider part 1, and for this first consider the case where $diag(R)= 1_N$, the "all ones vector", and $|v_1| > |v_2| > ... > |v_N|$.  Let $\{e_1,...,e_N\}$ be the natural basis, $q_i= (v_i/v_1 )e_1 + \sqrt{1-(v_i/v_1)^2} e_i$, $1\leq i\leq N$, $Q_0= [q_1, q_2, ..., q_N]$, $R_0= Q_0'Q_0$.  Note that by construction $||q_i||= 1$, $Q_0$ is lower triangular with positive diagonal entries, and thus is non-singular.  Also note, $e_i'Q_0e_1= v_i/v_1$, so $Q_0^{-T}v= v_1e_1$, and thus, $v'R^{-1}v= v'Q_0^{-1}Q_0^{-T}v= ||Q_0^{-T}v||^2= ||v_1e_1||^2= v_1^2= ||v||_\infty^2$.  But given any non-singular $R$ with $R=R'$, and $diag(R)=1_N$, there exists a Cholesky factorization, $R= Q'Q$, and by the Cauchy-Schwartz Lemma, $v'R^{-1}v= ||Q^{-T}v||^2 >= (q_1'Q^{-T}v)^2=(e_1'Q^T Q^{-T}v)^2= v_1^2= ||v||_\infty^2$; so, $R_0$ is a global minimum; and $min\{v'R^{-1}v\}= ||v||_\infty^2$.

Now consider the case where $diag(R)= 1_N$, and $v$ sorted, by $v$'s entries are not distinct.  The $Q_0$ as constructed above is singular, but since $v'R^{-1}v$ is a continuous function of $R$, it can be extended to $inf{v'R^{-1}v}= ||v||_\infty^2$.

Finally, consider the more general case.  Let $\Lambda= diag(r)^{-1/2}$, and $w= \Lambda v$.  Also, for each $R$, let $G= \Lambda R \Lambda$, so $diag(G)= 1_N$.  Then, $inf{v'R^{-1}v}= inf{v'\Lambda \Lambda^{-1} R^{-1} \Lambda^{-1} \Lamba v}= inf{w'G^{-1}w}= ||w||_\infty^2= = ||diag(r)^{-1/2}v||_\Inf^2.  This proves inf{v'*inv(R)v}= ||diag(r)^{-1/2}v||_\infty^2.

Now consider the second problem.  Let w= diag(r)^(1/2)sign(v), and Ro= ww'.  So, v'Ro v= sum{v_i v_j R[i,j]}= sum{v_i v_j \sqrt(r_i) sign(v_i) \sqrt(r_j) sign(v_j)}= ||diag(r)^(1/2)v||_1^2.  But |sum{v_i v_j R[i,j]}|<= sum{|v_i v_j R[i,j]|}<=sum{|v_i v_j \sqrt(r_i) \sqrt(r_j)|}= ||diag(r)^(1/2)v||_1^2; so, Ro is a global maximizer.  This proves sup{v'Rv}= ||diag(r)^(1/2)v||_1^2.  

Juxtaposing our solutions for parts 1 and 2, we have the following.
inf{v'*inv(R)v}= ||diag(r)^(-1/2)v||_\Inf^2
sup{v'Rv}= ||diag(r)^(1/2)v||_1^2
which exhibits a shocking duality.  Is this a coincidence, or a hint at something deeper?