Good, the oxygen is the other nucleophile. You're exactly right, the oxygen lone pair will attack the proton as the first step of the mechanism, but it won't form an alcohol straight out. The lone pair does the attacking, not the electrons from one of the carbon-oxygen bonds. What will form, though, is an epoxide, where the oxygen atom is still attached to both carbon atoms along with a proton. What is special about this type of oxygen atom? How might that modulate the reactivity of the epoxide. That is, after the first step where you have correctly indicated the oxygen atom attacks a proton, what are the new nucleophile(s)? what are the new electrophile(s)? Nucleophiles attack electrophiles.
As per the bonus, why is cis-2,3-dimethyloxirane not a chiral molecule? what is unique about the enantiomer of cis-2,3-dimethyloxirane?