[1] Can I do MBBS after a double degree [Bachelor of Engineering and Biomedical Science]?
Monash no longer offers the MBBS, which has been replaced by the MD (in general, they both qualify you to become a doctor). To answer your question - you cannot complete the MD at Monash via the undergraduate pathway if you have already begun tertiary study. However, your (Monash) Biomedical Science degree will allow you to apply for the graduate pathway (75 places).
[2] Does completing a double degree [Bachelor of Engineering and Biomedical Science] allow me to enter medical school and complete a M.D. degree and then [MAYBE] specialize to become a neurologist or a neurosurgeon?
Yes. In fact, in general you do not specifically require either degree to enter medical school in Australia, although Biomedical Science (at Monash) will allow you to apply for the graduate pathway into MD at Monash.
[3] Can I work as a starter engineer at a company while studying M.D. after school time?
I guess you could, but you probably wouldn't want to, given that you'll also have to balance the workload of studying medicine.
[4] Is it better to spend residency time here in Australia or outside the country?
I don't really have the knowledge to answer this, but hopefully someone else can.
Hope that helps