Is $a>b\iff \neg(b\geq a)$ true in constructive maths? 
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*Is $a>b\iff \neg(b\geq a)\;$ true in constructive maths? Why (not)?

*Also: is $\neg(a > b) \iff b\geq a\;$ true in constructive maths? Why (not)?

(Disclamer: This is based on what I remember about counterexamples involving creative subjects in intuitionism and my memory is a bit rusty. I may have made a mistake.)
I'm pretty sure that (1) is not true in intuitionism. Here is why:
Let $P$ be a mathematical proposition which has not yet been proved or disproved. Let $a$ be the decimal number $0.a_1a_2a_3\ldots$ where $a_i$ is 1 if $P$ has been proved or disproved at day number $i$ counting from today, while $a_i$ is $0$ if $P$ has still not been proved or disproved on that day. Let $b$ be $0$.
$b\geq a$ is absurd, as proving that would amount to proving that $P$ cannot be proved or disproved, which cannot be done in intuitionism (it's very different from an axiomatic system). Hence $\neg(b\geq a)$.
On the other hand, $a>b$ is not true (at present) because we cannot indicate a day in the future when $P$ will be proved.
Therefore, (1) is not true (at present).
