Dear Me and Minnie Me.
You should probably get a name for your child- haha
OK that aside.
Firstly, in some sense japan is one of the most friendly countries for the single parent. BUT to take advantage of the social services you must be on a proper visa (so any advice to the contrary is not only illegal but also silly).
Dependent Visa: As Glenski said, dependent visas for students are quite common, and provided that you are at a school that qualifies you for a visa, the dependent visa should not be a problem as long as you can demonstrate that you can support yourself and you dependent.
Day care: For single parents, you are first priority to get public day care which in my experience is GREAT! As a single parent, you will likely pay no fees.
Medical: Rules vary slightly according to the municipality but medical coverage up to Grade 6 is usually free and for single parents up to the end of high school (others pay 30% of the cost)
Choosing a school: There are literally TONS of language schools, so I would start by thinking where you want to live. (There will be a school there.)
Big city? advantages more international, English more widely used, all the big city stuff.
Small city? nursery school close to language school- can live with a bike, cheaper rent, etc etc.
I would suggest looking at municipal web pages, ones that have English webpages will probably respond to your email. Get the google map of the nursery schools (usually on the pages) and look at rent costs in that area and fro nearby language schools. Ask what your chances of getting in are (in big cities you might not get in all (to the nursery school). if you do not apply at the "right" time. But in smaller areas you can get in easily. Have done the nursery school thing, having it close to school and where you live is a big plus and in my opinion worth choosing a city based upon that.
I would really spend some time on google looking at where you might actually want to live, and put your child in daycare, and then last find a school.
Public Nursery school is very good and your child (AKA minnie me would learn Japanese at nursery school). If you stayed longer you might consider some kind of special class to help with Japanese, but I would not worry about it initially as there are many options after you arrive.
Hope that helps