The answer to both your questions is positive. But I totally disagree with your claim that minimal logic and intuitionistic logic are equivalent. To explain my viewpoint with an analogy, it is not because classical logic can be embedded in intuitionistic logic (via negative translation) that intuitionistic logic and classical logic are equivalent. Unless you redefine in a unnatural way the notion of "equivalence".
I answer your technical questions.
Suppose $\Gamma$ is a list of formulae and $A$ is a formula, all using just the logical symbols $\to,\lor,\land$. Suppose further that $\Gamma \vdash A$ is a theorem of intuitionistic logic. Is $\Gamma \vdash A$ a theorem of minimal logic?
Yes! If $\Gamma \vdash A$ is provable in intuitionistic logic, then there is a normal derivation $\pi$ of $\Gamma \vdash A$ in intuitionistic natural deduction, i.e. $\pi$ posses the subformula property. This entails that $\pi$ contains only inference rules for $\to, \lor, \land$, which are sound in minimal logic. Therefore, $\pi$ is a derivation in minimal natural deduction, i.e. $\Gamma \vdash A$ is provable in minimal logic.
So, all formulas provable in intuitionistic logic but not in minimal logic contain $\bot$ or $\lnot$ (remember that $\lnot A := A \to \bot$).
Suppose we add a bottom element to minimal logic; i.e. we add a new constant symbol $\bot_I$ and the rule $\bot_I⊢P$. Is this intuitionistic logic?
Yes! This is exactly the difference between intuitionistic logic and minimal logic. More precisely, they have the same language (with bottom $\bot$) and any derivation system for intuitionistic logic is defined as a derivation system for minimal logic plus the inference rule (that allows us to derive) $\bot \vdash P$ for every formula $P$ (the so called principe of explosion or ex falso quodlibet). Note that in minimal logic the bottom $\bot$ is nothing but a propositional variable, since in minimal logic there are no special inference rules for $\bot$ (there is no point in having a minimal bottom $\bot_M$ and an intuitionistic bottom $\bot_I$): $\bot$ in minimal logic does not have the same properties as in intuitionistic logic (as well as $\lor$ in intuitionistic logic does not have the same meaning as in classical logic).
Minimal logic can also be formulated in a language without $\bot$ but with negation $\lnot$ as primitive, adding (an inference rule that allows us to derive) the axiom $((A \to B) \land (A \to \lnot B)) \to \lnot A$ (see here, thanks to Peter Smith who advised me about this reference in his answer to a question of mine), but it is essentially equivalent to the formulation of minimal logic with $\bot$ and without the principle of explosion.
