"I am going to St. Petersburg."

If you're indicating motion towards something, you use the accusative case. "В петербург"

First is by some mode of transport, like a car or train, the second indicates by foot. But native Russians can confirm or deny this statememt of mine...

I was wondering the same thing.

And "Петер" is wrong as well, I think we should be given some more leniancy. What do you think ?

That's right, it's not.

ходить is used for walking/saying that you usually go to that place. Here, you're saying that you're going to Petersburg this time, so it's used идти