Although I'm majoring in Bioengineering systems, I haven't finished my degree (BBmed) so take this with a grain of salt.
1) You're mostly right. The bachelor degrees at UoM are designed to be quite broad to give you a range of expertise. Majoring in an engineering field in a bachelor degree doesn't really give you the 'full' qualifications to become an engineer, and so you are generally forced to do a Masters in Engineering. I don't have first-hand experience about whether doing a Masters gives you enough experience, but I do know that in your Masters, you will be doing semester- or year-long design projects. Becoming a fully accredited engineer from unimelb takes between 5 and 6 years (for most people, 5 years).
2) I would argue that UoM is one of the best universities for being able to explore other interests. The Bachelor of Science is a really flexible course, and along side doing engineering subjects, you can explore other areas such as biology, chemistry, physics, etc. As you mentioned, breadth subjects force you to take subjects that are outside your field. In your case, that could be arts, commerce, music (really anything you want other than science). You have to take between 4 to 6 breadth subjects, meaning this is a great way to explore your interests in other fields. There are also concurrent diplomas (music, languages, computing and mathematical sciences), which although usually add an extra year, give you an extra qualification.
As you can see in my signature, I've pretty much fully customised my degree. I'm taking subjects in biology and chemistry as core units; physics, computing and engineering subjects for my major; maths for my concurrent diploma; and for breadth, languages, music and commerce.