As for your first question, concerning nonaffine R-varieties as you call them, yes, there are nonaffine R-varieties. However, they are considered pathological. Example 12.1.5 on page 301 of Bochnak-Coste-Roy, Real algebraic geometry, constructs an R-line bundle over $\mathbf R^2$ whose total space is not affine. In fact, it is not affine since it does not have any separated complexification. Note that the R-variety itself, however, is separated!

The essential point here is that the set of real points of an irreducible affine scheme over $\mathbf R$ can be reducible. In the aforementioned example, the irreducible scheme in question is the one defined by the irreducible polynomial $$p=x^2(x-1)^2+y^2\in\mathbf R[x,y,z].$$ The set of real points in $\mathbf R^3$ defined by $p$ is the disjoint union of the affine lines $$L_0=\{(0,0)\}\times\mathbf R\ \mathrm{and}\ L_1=\{(1,0)\}\times\mathbf R. $$ This is clearly a reducible subset of $\mathbf R^3$. The separated R-variety that does not have a separated complexification is the one obtained by gluing the open subsets $$ U_0=\mathbf R^3\setminus L_0\ \mathrm{and}\ U_1=\mathbf R^3\setminus L_1 $$ along the open subsets $$ U_{01}=U_0\cap U_1\subseteq U_0\ \mathrm{and}\ U_{10}=U_0\cap U_1\subseteq U_1 $$ via the regular isomorphism $$ \phi_{10}\colon U_{01}\rightarrow U_{10} $$ defined by $$ \phi_{10}(x,y,z)=(x,y,pz). $$ Note that this is indeed a regular isomorphism since the map $\phi_{01}=\phi_{10}^{-1}$ is the regular map $$ \phi_{01}\colon U_{10}\rightarrow U_{01} $$ defined by $$ \phi_{01}(x,y,z)=(x,y,\tfrac{z}{p}). $$

Now, it is easy to see that the R-variety $U$ one obtains is separated, as defined in the founding paper of the whole theory: Faisceaux algébriques cohérents by Jean-Pierre Serre. Indeed, one easily checks that the diagonal in $U\times U$ is closed. However, if one wants to construct a real scheme $X$ whose set of real points coincides with $U$, then, inevitably, $X$ will not be separated. Indeed, the polynomial $p$ defines a nonclosed point $x_0$ in any scheme-wise thickening $X_0$ of $U_0$ since $p$ has zeros in $U_0$, and similarly it defines a non closed point $x_1$ of any scheme-wise thickening $X_1$ of $U_1$. The gluing morphisms $\phi_{01}$ and $\phi_{10}$ will extend to open subsets $X_{01}$ of $X_0$ and $X_{10}$ of $X_1$, but they won't contain $x_0$ and $x_1$, respectively. This is because the polynomial $p$ vanishes at $x_0$. As a result, any scheme-wise thickening of $U$ will be nonseparated!

As for your second question, if I understand correctly, you are asking whether the functor $$ F\colon Sch_R'\rightarrow R-Var $$ defined by $F(X)=X(\mathbf R)$ is an equivalence onto a full subcategory, where $Sch_R'$ is the category of finite type separated reduced schemes over $Spec(\mathbf R)$ having dense sets of real points. Indeed, this is an equivalence onto a full subcategory, its image category, since any morphism of $R$-varieties wil extend to a morphims defined on some open subset containing the real points. Uniqueness is implied by density of real points and separation.

As for your third question, I can't think of other differences between $R$-varieties and schemes over $\mathbf R$ that differ essentially from phenomena already present in the example above.

As for your final question about varieties in the sense of $R$-varieties over other fields, Serre certainly did define them in the paper I mentioned above. I'm not sure whether that has had a follow-up for other fields than real or algebraically closed fields.

As for your first question, concerning nonaffine R-varieties as you call them, yes, there are nonaffine R-varieties. However, they are considered pathological. Example 12.1.5 on page 301 of Bochnak-Coste-Roy, Real algebraic geometry, constructs an R-line bundle over $\mathbf R^2$ whose total space is not affine. In fact, it is not affine since it does not have any separated complexification. Note that the R-variety itself, however, is separated!

The essential point here is that the set of real points of an irreducible affine scheme over $\mathbf R$ can be reducible. In the aforementioned example, the irreducible scheme in question is the one defined by the irreducible polynomial $$p=x^2(x-1)^2+y^2\in\mathbf R[x,y,z].$$ The set of real points in $\mathbf R^3$ defined by $p$ is the disjoint union of the affine lines $$L_0=\{(0,0)\}\times\mathbf R\ \mathrm{and}\ L_1=\{(1,0)\}\times\mathbf R. $$ This is clearly a reducible subset of $\mathbf R^3$. The separated R-variety that does not have a separated complexification is the one obtained by gluing the open subsets $$ U_0=\mathbf R^3\setminus L_0\ \mathrm{and}\ U_1=\mathbf R^3\setminus L_1 $$ along the open subsets $$ U_{01}=U_0\cap U_1\subseteq U_0\ \mathrm{and}\ U_{10}=U_0\cap U_1\subseteq U_1 $$ via the regular isomorphism $$ \phi_{10}\colon U_{01}\rightarrow U_{10} $$ defined by $$ \phi_{10}(x,y,z)=(x,y,pz). $$ Note that this is indeed a regular isomorphism since the map $\phi_{01}=\phi_{10}^{-1}$ is the regular map $$ \phi_{01}\colon U_{10}\rightarrow U_{01} $$ defined by $$ \phi_{01}(x,y,z)=(x,y,\tfrac{z}{p}). $$

Now, it is easy to see that the R-variety $U$ one obtains is separated, as defined in the founding paper of the whole theory: Faisceaux algébriques cohérents by Jean-Pierre Serre. Indeed, one easily checks that the diagonal in $U\times U$ is closed. However, if one wants to construct a real scheme $X$ whose set of real points coincides with $U$, then, inevitably, $X$ will not be separated. Indeed, the polynomial $p$ defines a nonclosed point $x_0$ in any scheme-wise thickening $X_0$ of $U_0$ since $p$ has zeros in $U_0$, and similarly it defines a non closed point $x_1$ of any scheme-wise thickening $X_1$ of $U_1$. The gluing morphisms $\phi_{01}$ and $\phi_{10}$ will extend to open subsets $X_{01}$ of $X_0$ and $X_{10}$ of $X_1$, but they won't contain $x_0$ and $x_1$, respectively. This is because the polynomial $p$ vanishes at $x_0$. As a result, any scheme-wise thickening of $U$ will be nonseparated!

As for your second question, if I understand correctly, you are asking whether the functor $$ F\colon Sch_R'\rightarrow R-Var $$ defined by $F(X)=X(\mathbf R)$ is an equivalence onto a full subcategory, where $Sch_R'$ is the category of finite type separated reduced schemes over $Spec(\mathbf R)$ having dense sets of real points. This is an equivalence onto a full subcategory, its image category, if you localize $Sch_R'$ with respect to inlcusions of open subsets containing all real points: any morphism of $R$-varieties wil extend to a morphims defined on some open subset containing the real points. Uniqueness is implied by density of real points and separation.

As for your third question, I can't think of other differences between $R$-varieties and schemes over $\mathbf R$ that differ essentially from phenomena already present in the example above.

As for your final question about varieties in the sense of $R$-varieties over other fields, Serre certainly did define them in the paper I mentioned above. I'm not sure whether that has had a follow-up for other fields than real or algebraically closed fields.