Step 1: Recall why beryllium behaves anomalously.
Beryllium's very small ionic size gives $\text{Be}^{2+}$ unusually high charge density, so its compounds often break the pattern set by the heavier alkaline earth metals.
Step 2: Check statements I and III.
Statement I, that beryllium halides are covalent, is correct, since the small, highly polarising $\text{Be}^{2+}$ distorts the halide ion's electron cloud strongly. Statement III, that $\text{BeCO}_3$ is unstable, is also correct, it decomposes readily and must be stored under a $\text{CO}_2$ atmosphere.
Step 3: Check statements II and IV.
Statement II is wrong because $\text{Be(OH)}_2$ is amphoteric, dissolving in both acids and strong alkalis, unlike the basic hydroxides of the heavier members. Statement IV is wrong because BeO actually adopts the wurtzite structure due to beryllium's small size and covalent character, not the rock salt structure seen in MgO or CaO.
Final answer: Option 4, statements II and IV are the incorrect ones.