Help needed with calendar and reminder alerts with do not disturb on IPhone 5C

Hi everyone, this is my first post here. I just got my first IOS device, an iPhone 5C, the Friday before last, and it's a blast. I'm having a problem with calendar and reminder alerts, though. I'm trying to get reminders and calendar alerts to at least show up in the home screen with do not disturb on like missed phone calls and text messages do, but I'm not having much luck. I tried turning push on and setting the fetch time in Settings\Mail, Contacts, and Calendars, with no result. I also tried changing things around in Settings\Notification Center, but still nothing. Somehow, one of my reminders showed up in the home screen yesterday, but I haven't been able to duplicate that since. Calendar alerts do not show up there at all. Is there something I'm doing wrong, or is there an app that fixes this? Is there an app that would allow calendar and/or reminder alerts to sound regardless of whether do not disturb is on or not, like the alarm in the clock? I don't want to be awakened by texts and calls when I'm trying to sleep, but I need my calendar and reminder alerts to at least show up in the home screen like my missed calls and texts do. Can anyone please help?