Hi.

When you say "register" does this mean registering as a business? Does this mean I can charge more if I do?
No. Registering a business and registering a day home are two different terms. First off - you can register your business if you intend having a business name vs operating under your own name. Most province's require this. i.e. I can run a day home that doesn't have an actual business name and where I advertise under my own name and I don't have to register my business. But if I want to call my business "Little Bees" or anything else, then the business name must be registered in my province. They check first that no one else has that business name (there's a fee for checking). If the business name is already taken, then I'd have to pay again for my next choice of business name. Once I've found a name that no one else is using, then I pay more to reserve it exclusively for me and for the use of the business name.

Being a registered day home normally means, being licensed by the Department of Education (or whatever it's called in your province), it normally means working with a third party (an agency) who are tasked with ensuring you are meeting a certain level of care in your program.

The rules about working with an agency, having a license, working independently, vary from province to province as child care is legislated on a provincial level. Likewise the terminology varies slightly too.

Do dayhomes charge GST?
No. There's a couple of good CRA guides on line. One is called "Using your home for day care" and the other is called "self employment" (or something similar). Information about all topics like this is normally covered between the two publications.

What other legalities must you have to open a dayhome?
That depends on what province you are in, and what type of day home you will have. Where are you located?

What is the initial investment on this?
Again, that depends on what your provincial legislation requires and if your home meets those needs without changes or not. I would strongly suggest day home insurance because even if your province doesn't require it, you will find many house insurance policies won't pay out in the event of a claim if they didn't know you were running a business from home. Likewise, I'd strongly suggest police checks and child abuse register checks plus first aid training. Not all provinces require this either but it's reassuring for parents to know these checks are available for them to see. Other than that - cribs/cots/toys/craft supplies/bedding/high chairs - all the normal supplies you'd expect a child care operation to have on hand.

Can I hire an employee to do the work? (and still keep my day job)
For most provinces - no. A day home is typically run by the primary resident of the house who is the person insured to have a day home on their premises.
And would you really want a member of staff, who you will have to pay at least min wage, in your personal home, if you were out at work, caring for children. To be honest, anyone who would take that job would likely open their own day home.

Again your other questions are dependent on where you are. But it sounds to me like you have a lot of research to do first.