Yes, the language is recognized by a co-Turing-recognizable machine.
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No, it is not possible to show that the language recognized by an infinite pushdown automaton is decidable.
Machine Language
Yes, the difference between decidable and recognizable languages in theoretical computer science is clear to me. Decidable languages can be recognized by a Turing machine that always halts and gives a definite answer, while recognizable languages can be recognized by a Turing machine that may not always halt, but will give a positive answer for strings in the language.
A Turing machine can be built to accept the language defined by the keyword.
Non-Turing recognizable languages are languages that cannot be recognized by a Turing machine. Examples include the language of palindromes over a binary alphabet and the language of balanced parentheses. These languages differ from Turing recognizable languages in that there is no algorithmic procedure that can determine whether a given input belongs to the language.