There is no such thing as an "Islamic" calendar. Only thing the Quran stated was that there were 12 months and four were special or however one wants to translate it. The calendar months today are based on Arab tradition in East Arabia. The Quran was speaking to an Arabian people in the 7th century. From archeological findings, each region in Arabia had their own calendar. Read this https//en.wikipedia.org/wiki/Pre-Islamic_Arabian_calendar . Whether the calendar is lunar, solar, lunisolar... it does not matter. As for the crescent moon, it is not a symbol of Islam. It goes way back to Akkadian who used the crescent symbol to represent their god Syn.