In British English, ‘like+ -ing form is used mostly to talk about enjoyment/pleasure, and ‘like+infinitive’ mostly to talk about choices and habits. In American English, ‘like+infinitive’ is common in both senses. 'I like to keep all these books in order.' (it's a good idea) 'I like reading books.' (a pleasure)

Same as vanilla ice cream and plain ice cream.

same

YES! both are the same. You can say 'She likes verb+ing' or 'She likes to verb'. So you can say 'I like studying English' or 'I like to study English'. Hope this helps :)