Metals definitely don't have low boiling points. If you mistook this for low melting point, then I must tell you only alkali metals (group 1) exihibit low melting point.
Group VII elements occur as diatomic molecules naturally. Their intermolecular bonding is Van Der Waal's in nature. Their boiling point will definitely increase with molecular weight (as it means these molecules also contain more electrons, hence stronger intermolecular bonding).
Most non-metals have low melting point, with reference to their structures.
Melting point of transition metals increases down the periodic table because there are more delocalised electrons, consequently stronger metallic bond. Hence, the increase in melting point.
Only the 2nd statement is incorrect.