For the first question you already have had an answer in http://mathoverflow.net/questions/25687/projective-module/25698 if $\mathrm{Ext}^1_{\mathbb Z}(P,M)=0$, then it depends on the axioms of set-theory whether the conclusion is true or not. The answer to the second question is yes, it is one of the basic characterisation of injective modules that $\mathrm{Ext}^1(A/I,Q)=0$ for all ideals $I$ iff $Q$ is injective. As for the question in your title, the answer should be no for the second variable (irrespective of the axioms of set theory, but I am too lazy to try to come up with an example). For the first variable things are a little bit more interesting: If $M$ is the direct limit of ${M_\alpha}$, then we have a spectral sequence with $E_2$-term $\varprojlim^i\mathrm{Ext}^j(M_{\alpha},Q)$ lim^i\mathrm{Ext}^j(M_{\alpha},Q)$("lim" means inverse limit, there is some strange problem with using "varprojlim" which sometimes works and sometimes doesn't) and converging to$\mathrm{Ext}^{i+j}(M,Q)$. Somewhat strangely this spectral sequence does not seem to formally give the above characterisation of injective modules as there is a potential$\varprojlim^1\mathrm{Hom}(M_{\alpha},Q)$lim^1\mathrm{Hom}(M_{\alpha},Q)$ contribution.
For the first question you already have had an answer in http://mathoverflow.net/questions/25687/projective-module/25698 if $\mathrm{Ext}^1_{\mathbb Z}(P,M)=0$, then it depends on the axioms of set-theory whether the conclusion is true or not. The answer to the second question is yes, it is one of the basic characterisation of injective modules that $\mathrm{Ext}^1(A/I,Q)=0$ for all ideals $I$ iff $Q$ is injective. As for the question in your title, the answer should be no for the second variable (irrespective of the axioms of set theory, but I am too lazy to try to come up with an example). For the first variable things are a little bit more interesting: If $M$ is the direct limit of ${M_\alpha}$, then we have a spectral sequence with $E_2$-term $\varprojlim^i\mathrm{Ext}^j(M_\alpha,Q)$ \varprojlim^i\mathrm{Ext}^j(M_{\alpha},Q)$and converging to$\mathrm{Ext}^{i+j}(M,Q)$. Somewhat strangely this spectral sequence does not seem to formally give the above characterisation of injective modules as there is a potential$\varprojlim^1\mathrm{Hom}(M_\alpha,Q)$\varprojlim^1\mathrm{Hom}(M_{\alpha},Q)$ contribution.
For the first question you already have had an answer in http://mathoverflow.net/questions/25687/projective-module/25698 if $\mathrm{Ext}^1_{\mathbb Z}(P,M)=0$, then it depends on the axioms of set-theory whether the conclusion is true or not. The answer to the second question is yes, it is one of the basic characterisation of injective modules that $\mathrm{Ext}^1(A/I,Q)=0$ for all ideals $I$ iff $Q$ is injective. As for the question in your title, the answer should be no for the second variable (irrespective of the axioms of set theory, but I am too lazy to try to come up with an example). For the first variable things are a little bit more interesting: If $M$ is the direct limit of ${M_\alpha}$, then we have a spectral sequence with $E_2$-term $\varprojlim^i\mathrm{Ext}^j(M_\alpha,Q)$ and converging to $\mathrm{Ext}^{i+j}(M,Q)$. Somewhat strangely this spectral sequence does not seem to formally give the above characterisation of injective modules as there is a potential $\varprojlim^1\mathrm{Hom}(M_\alpha,Q)$ contribution.