Do I need FileMaker Server (which I do not have)?
FMS is not required for development. Depends on the number of users / concurrent users that need to access the database, and if webdirect is required. Over 5 concurrent users and or webdirect will require FMS.
Can I design something with FMPA15 and actually get it into the stores?
You can get database into app store using the iosAppSDK. There is nothing available for the Apple Store. I'm sure this is coming in the near feature. I believe since Runtimes have been deprecated, that is FMI next move.
Can I create a community module, whereby our patrons update their own profiles?
Yes. This move would most likely require you to have FMS.
Will reminders and announcements integrate with the Notification component of users' phones?
Not, build into FM or iosAppSDK, but MoneyBreadSoftware has an sdk and there are a couple other software apps available to handle the communication from the app.
I'm not sure all what MoneyBreadSoftware offers you would need to check their website.