Yes, several.
No. That's not what it means. Neoliberal is a term that refers to ideas going all the way back to classical liberalism. Namely, free trade, free markets, very limited government interference, etc you get the point. Neoliberal is sort of a misleading term because these ideas have been around for a while, but they've been dug up recently and given a modern flair, but nobody really would identify as one in modern times. Ironically in your argument's case, the Democratic Party under Jackson was probably the biggest proponent of these ideas.
No I'm not, I'm quite aware of that.
Er. Not really. Jackson certainly ran as a populist, he definitely liked to hate on the banks, the privileged and concentrated wealth but his economic policies are obviously no way similar to the modern Democratic Party.