Is there a way to get Google Calendar reminders to act like Alarms? I miss many calendar reminders because they're just too subtle. I might put my phone down and walk out of the room for 5 minutes, missing the reminder, then come back and not touch my phone again for an hour; completely missing an important appointment.

If reminders worked like alarms, they would make their notification sound continuously (up to a point, like alarms: ~10 minutes), and take over the whole screen, until silenced. That would be preferable to me. Is there an app that can do this?
If you want to stick with your Google Calendar as backend and frontend, you might wish to check Calendar Event Reminder. You can tell that app which calendars to look at, and to annoy you until you're going crazy. I almost guarantee you cannot miss such a reminder! I've used the app for quite some time, before...

I switched to a different calendar frontend: Business Calendar. Very configurable again, also concerning reminders: vibrates and makes "subtle noises" (I've told him so) until I lose my nerves and switch it off :)

There are probably a lot of similar approaches. But I know of no way to even come near to one of them without any 3rd party tool being involved.