I can't answer the first question regarding the Levamentum Project, but generally a tank style binder comes to around the ribcage or belly button while a full length binder can be pulled over the hips. Both will compress your chest down to a more masculine level, it's a matter of if you want your hips covered or not. Mostly a comfort thing.
I'm going to assume since you're asking if the doctor will explain to your parents about your feelings that you're underage. Depending on the circumstances, the doctor will probably not do the talking for you. He/she may offer to be there when you tell your parents, offer to explain about gender dysphoria, or give you guidance on how to talk to your parents. Though if your parents are paying for you to see the psychologist or therapist, they'll likely want to know why you're picking one that specializes in gender identity disorders. But definitely call a therapist first. I don't know about all the countries besides the US, but most doctors who can do SRS or HRT will require a letter of recommendation before seeing you.
And it's common for transmen to feel disappointed about their penises. I know I am, but it seems worse than it is. Surgeons are looking to improve surgeries all the time. While it's true that a phallo will never allow for a natural erection(until they figure out how to replicate the tissue in a penis and build them from scratch for us) but you will be able to get erect and use your new penis for penetration. It won't be any different than a cis man who has erectile dysfunction or a penis injury and requires a pump in order to get hard.
Hope that helps.
