We can associate a logic to an algebra in the following way:
Let $(A, \leq)$ be an algebra and $D \subseteq A$ a set of designated elements. Then we define a valuation as $v: Prop \to A$. Then we can define the propositional logic of $(A, \leq)$ as $L(A,D)= \{ \varphi : v(\varphi)\in D $ for any valuation $v \}$. Now in the case of Boolean algebras and classical propositional logic we have the follwoing theorem.
For any Boolean algebra $B$ we have $L(B, \{ 1\})=$ Classical prop. logic.
1.) What happens in the case of heyting algebras H ? Do we get $L(H, \{ 1\})=$ Intuitionistic prop. logic. ? Or do we get some Intermediate logic in most cases ?
2.) Does the logic associated to the three valued Heyting algebra with $\{ 1\}$ as designated element validate the weak law of excludded middle ?
Regarding (1) I think in many cases you get intermediate logics. Consider for instance, the three-valued Heyting algebra $H_3$, then $L (H_3, \{1\}) = IPL + (p \to q) \vee (q \to p )$.
Am I missing something ?
You certainly will not always get intuitionistic logic. For instance, if $H$ is actually a (nontrivial) Boolean algebra, then you get Boolean logic. And if $H$ is the trivial (1-element) algebra you get something even stronger than Boolean logic in which everything is true. Or if $H$ is a chain with more than one element, then the logic you get will satisfy the weak law of excluded middle but not the law of excluded middle since the negation of every element is either $0$ or $1$.
In general, the different logics you get correspond to different sets of identities that can be true in a Heyting algebra, or in the language of universal algebra, the subvarieties of the variety of Heyting algebras. The case of Boolean algebras happens to be kind of special because they actually have no subvarieties besides the whole variety and just the trivial algebra, so all nontrivial Boolean algebras give the same logic. But for Heyting algebras there is much more variation.