A Jew who sins remains a Jew (Sanhedrin 44a). Rav Soloveichik, in Five Derashot explains that there are two aspects to Jewishness: the national aspect (haberit ben habeitarim), which is not connected to free choice and the religious aspect (berit Horev), which is. In fact, we see that a secular Jew today will scream if someone contends that he is not Jewish.
As for the O.P.’s question, Rav Kook explains (Al HaKefira that kefira has its source in some lack among observant Jews. In fact, the Chafetz Chaim once commented that if a Jew in Vilna puts down his Gemara to early a Jew in Paris (then a secular bastion) will eat pork. Thus there is areivut.
As for children of non-Jewish mothers, while they are not halachically Jewish some mekubbalim say that there is an inyan to bring them back. The work of Rav Eliahu Avichail and of Michael Freund’s “Shavei Yisrael” is well-known.