I believe the following are all true, and I could probably prove them myself if necessary. However this would be inelegant, and I would much prefer just to have references. I have been Googling quite a bit but no luck.
Let $E$ be a Lebesgue-measurable subset of $\mathbb{R}$, and let $\mathcal{B}=\sigma(\tau)$ be the Borel $\sigma$-algebra on the usual metric topology $\tau$ for $\mathbb{R}$. Denote by $E\cap\mathcal{B}$ the trace of $\mathcal{B}$, i.e. $$E\cap\mathcal{B}=\{E\cap B:B\in\mathcal{B}\}.$$ We also denote by $\tau_E$ the subspace topology on $E$, i.e. $$\tau_E=\{E\cap U:U\in\tau\}.$$
According to proofwiki, the following is true:
(1) $\sigma(\tau_E)=E\cap\mathcal{B}$. In other words, the Borel $\sigma$-algebra on $(E,\tau_E)$ is just the family of all subsets of $E$ of the form $E\cap B$ for some Borel $B\subseteq\mathbb{R}$.
However, I need a "real" reference, i.e. something from a reputable publication.
Let $F$ be another Lebesgue-measurable subset of $\mathbb{R}$, and suppose $f:F\to E$ is a bijective function which is strictly increasing. Then it is measurable in the following sense: If $c\in\mathbb{R}$ then $\{x\in F:f(x)<c\}\in F\cap\mathcal{B}$.
(2) Is it true that $f^{-1}(A)\in F\cap\mathcal{B}$ for all $A\in E\cap\mathcal{B}$? If (1) is true, this is equivalent to asking if $f^{-1}(A)\in\sigma(\tau_F)$ for all $A\in\sigma(\tau_E)$.
Finally, I have this one last reference request:
(3) Is it true that if $C$ is a Lebesgue-measurable subset of $\mathbb{R}$ then there is a Borel set $D\in\mathcal{B}$ such that $C\subseteq D$ and $D\setminus C$ has Lebesgue measure zero?
I have heard that we can find $G\in\mathcal{B}$ such that $C\Delta G$ (the symmetric difference) has Lebesgue measure zero, and also that for any Lebesgue measure-zero set there exists a Borel set $H\in\mathcal{B}$ of measure zero which contains it. If both of these facts are true then letting $D=G\cup H$ will do the trick for (3). But, again, I need references.
Thanks guys!
EDIT: I suppose if I can independently verify that these three things are all true, then I could just say something like "it is well-known that..." and see if the referee complains. But references would be ideal.