For your last Q. If $f:r\to R$ is increasing and $r\in R,$ then $f^{-1}(r,\infty)$ is $\phi, $ or is $R, $ or is $(\;\inf f^{-1}(r,\infty)\;), $ or is $[\min f^{-1}\{r\},\infty\;), $ which is in all cases a Borel set. Similarly, for $s\in R, $ the set $f^{-1}(-\infty,s)$ is Borel. So $f^{-1}(s,r)$ is Borel. Every open set of reals is a countable union of open intervals (see below); therefore $f^{-1}U$ is Borel for every open $U.$ All of this can be done in ZF.

Still in ZF , we have: $Q$ and $Q\times Q$ are countable and $Q$ is dense in $R.$ If $f,g$ are measurable and $s<r$ then $(f+g)^{-1}(s,r)=$ $\cup \{f^{-1}(x,y)\cap g^{-1}(a-x,b-y):x,y\in Q\}$ is a measurable set. As above, therefore $(f+g)^{-1}U$ is a measurable set for every open $U.$

To show in ZF that every open $U\subset R$ is a countable union of pair-wise disjoint open intervals (including possibly unbounded ones): For convenience let In $(x,y)=(x,y)\cup (y,x)$ for $x,y \in R.$ (In $(x,y)$ is the open interval between $x$ and $y.$) For any open $U\subset R$ and $x,y\in U$ let $x\equiv_U y\iff$ In$(x,y)\subset U.$ We easily show this is an equivalence relation on $U,$ and that each equivalence class $[x]_{\equiv_U}$ is an open interval. Now $U=\cup \{[x]_{\equiv_U} :x\in Q\cap V\}.$

For your last Q. If $f:r\to R$ is increasing and $r\in R,$ then $f^{-1}(r,\infty)$ is $\phi, $ or is $R, $ or is $(\;\inf f^{-1}(r,\infty)\;), $ or is $[\min f^{-1}\{r\},\infty\;), $ which is in all cases a Borel set. Similarly, for $s\in R, $ the set $f^{-1}(-\infty,s)$ is Borel. So $f^{-1}(s,r)$ is Borel. Every open set of reals is a countable union of open intervals (see below); therefore $f^{-1}U$ is Borel for every open $U.$ All of this can be done in ZF.

To show in ZF that every open $U\subset R$ is a countable union of pair-wise disjoint open intervals (including possibly unbounded ones): For convenience let In $(x,y)=(x,y)\cup (y,x)$ for $x,y \in R.$ (In $(x,y)$ is the open interval between $x$ and $y.$) For any open $U\subset R$ and $x,y\in U$ let $x\equiv_U y\iff$ In$(x,y)\subset U.$ We easily show this is an equivalence relation on $U,$ and that each equivalence class $[x]_{\equiv_U}$ is an open interval. Now $U=\cup \{[x]_{\equiv_U} :x\in Q\cap V\}.$

For your last Q. If $f:r\to R$ is increasing and $r\in R,$ then $f^{-1}(r,\infty)$ is $\phi, $ or is $R, $ or is $(\;\inf f^{-1}(r,\infty)\;), $ or is $[\min f^{-1}\{r\},\infty\;), $ which is in all cases a Borel set. Similarly, for $s\in R, $ the set $f^{-1}(-\infty,s)$ is Borel. So $f^{-1}(s,r)$ is Borel. Every open set of reals is a countable union of open intervals; therefore $f^{-1}U$ is Borel for every open $U.$ All of this can be done in ZF.

For your last Q. If $f:r\to R$ is increasing and $r\in R,$ then $f^{-1}(r,\infty)$ is $\phi, $ or is $R, $ or is $(\;\inf f^{-1}(r,\infty)\;), $ or is $[\min f^{-1}\{r\},\infty\;), $ which is in all cases a Borel set. Similarly, for $s\in R, $ the set $f^{-1}(-\infty,s)$ is Borel. So $f^{-1}(s,r)$ is Borel. Every open set of reals is a countable union of open intervals (see below); therefore $f^{-1}U$ is Borel for every open $U.$ All of this can be done in ZF.

Still in ZF , we have: $Q$ and $Q\times Q$ are countable and $Q$ is dense in $R.$ If $f,g$ are measurable and $s<r$ then $(f+g)^{-1}(s,r)=$ $\cup \{f^{-1}(x,y)\cap g^{-1}(a-x,b-y):x,y\in Q\}$ is a measurable set. As above, therefore $(f+g)^{-1}U$ is a measurable set for every open $U.$

To show in ZF that every open $U\subset R$ is a countable union of pair-wise disjoint open intervals (including possibly unbounded ones): For convenience let In $(x,y)=(x,y)\cup (y,x)$ for $x,y \in R.$ (In $(x,y)$ is the open interval between $x$ and $y.$) For any open $U\subset R$ and $x,y\in U$ let $x\equiv_U y\iff$ In$(x,y)\subset U.$ We easily show this is an equivalence relation on $U,$ and that each equivalence class $[x]_{\equiv_U}$ is an open interval. Now $U=\cup \{[x]_{\equiv_U} :x\in Q\cap V\}.$