Higher Education in the Wizarding World

My last post dealt with a super serious topic, so I thought I should post something lighthearted to balance it out. A few weeks ago or maybe it was months, I don't know because it all runs together, I was behind a car with a Harry Potter bumper sticker. You'd better believe I was all over that.

Obviously I found this amusing, but then I decided to spend a significant amount of time thinking about it (that's what I do). My over-thinking led to a number of questions:

• Does Hogwarts have tuition? They don't mention it in the book, but I'm sure it does. I mean, they provide room and board to all the students for crying out loud. It's a far cry from your traditional public school. 
• Are there scholarships? There must be, otherwise how could the Weasleys afford to send all of their children there? 
• Are there other school options in the wizarding world? You have to be accepted to Hogwarts based on your potential for magical ability but not all children born to wizard parents have that ability. Take Filch, for example, he's a Squib (non-magical person born to wizard parents). What would be the point of a Squib going to Hogwarts? 
• Where do wizard children go to school prior to the age of eleven? And what do they learn? Regular math and stuff?
• Did Muggle parents ever question their children being accepted? "Hey honey, an owl just dropped off this letter saying our kid is a witch/wizard and that we should send her/him to this school that we've never heard of. Well, that seems legit." Did any of them ever say no?

Technically FAFSA applies to post-secondary education and that's not what Hogwarts is. Hogwarts is a secondary school. If you want to get bogged down in semantics, this bumper sticker is inaccurate. But it brings me to my main quesstion: Why is there no higher education for wizards? I can't imagine that sat well with Hermione Granger.

Seven years of schooling and you're done. That's all the education you need as a wizard. You're 18 years old and nobody thinks you should maybe go to some more classes? Not even to do something really specialized, like be an Auror or a Professor? That means that by the end of their time at Hogwarts, Seamus Finnegan and Neville Longbottom had the same level of education as Albus Dumbledore. Sure he had decades of life experience, but they had taken all the same classes.

I'm required to have a Masters and all I teach is Spanish, I don't even get into levitation, transfiguration, Dark Arts, charms, etc. If I were going to send my kid off to learn all of that stuff I'd feel more at ease knowing that the teacher had more than the wizard equivalent of a high school diploma under his hat.

For the record: I would go to Hogwarts today if an owl showed up at my house. Same goes for my future children, I'm so down with them attending Hogwarts and learning magic.