They do now, but this appears to be due only to the cultural domination of Islam (examples of biblical texts using the word "Allah" cannot be found before around 900AD). Before Mohammed, "Allah" was the name of one of the deities of the Kaaba (a member of a pantheon of 360 gods), not another name for the Jewish God.
"Allah" means "God" in Arabic.
No it doesn't. "God" would be "Ilah", not "Allah", in arabic. "Allah" is a proper name of a specific deity, not a generic noun.
The article starting this thread suggests that "Allah" is from the compound Arabic word "al-ilah" or in english "the god". While this is popularly held to be true, in reality such an origin seems doubtful, since the part of the word signifying deity, "IL", is missing from the compound word. If this were the origin, then "Allah" would mean "the the", not "the god", which doesn't make sense. The Muslims believe that "Allah" is the proper name for their god, not a generic name for the Deity.