A lot of names have its counterpart in other languages. Czech Jan has a similar origin to John in English, Johann in German, Juan in Spanish. (its female counterpart in Czech is Jana). I believe that if you come to CR and introduce yourself as Matthew, nobody would call you Matěj, unless you specifically ask for it. Hope it makes sense.
As for the "English" question you are right - it translates as Angličan (masculine) and Angličanka (feminine); derived from Anglie (England).
Not really the answer to your question, but: a woman's surname usually (or, at least, often) gets the "-ová" ending. For example, https://cs.wikipedia.org/wiki/Mary_Shelleyov%C3%A1