People make way too much of the different ways "Shakespeare" was spelled. Hard as this is to imagine, there was simply wasn't fixed spelling like we have today; people spelled things different ways, based on how a word sounded to them, how they were used to seeing it, how they were using the word...people would even spell a word different ways in the same letter. The Oxford gang is really good at grabbing little "facts" from the historical record and completely misinterpreting them, creating rules that apply one way for Oxford and another way for Shakespeare. It's pointless and annoying, but I guess we're stuck with it. Everyone loves a good conspiracy theory.
But, back to names--Shakespeare's name was most often spelled just like that: Shakespeare (almost 3 times as often as any other spelling). The most common variations were Shakespear and Shakspere. If you think that means he wasn't a real person, or was two different people, consider that Edward de Vere, the Earl of Oxford, signed his name both "Oxford" and "Oxenford" all the time--so he must be two people, too!
As for school, Shakespeare went to a grammar school in Stratford, where, after learning his ABC's, he mostly studied Latin and Greek. Oxfordians like to paint him as some kind of illiterate country bumpkin, but, come on, people, his father was the Mayor of Stratford! He did OK. When you read his plays, you see a lot of classical references, but they keep coming from the same handfull of books--Ovid's Metamorphoses, Plutarch's Lives, that sort of thing. You want to guess which books he would have read in grammar school?