Can I efficiently use the iPhone with a prepaid Sim card?
This is perfectly doable. I use a prepaid SIM myself and get by without a problem (although my usage is a lot different to yours). Here are some tips to maximise your iPhone efficiency and minimise your bill.


Switch off "Background App Refresh". I have mine switched off for all but a few apps but for your needs it isn't necessary. This will boost battery life as well as reduce data usage since apps won't be able to update their content while backgrounded.
Set your mobile data network to 2G, and switch to 3G only when necessary. Again, this will boost your battery and reduce data costs, while still allowing Push notifications to come through.
Switch Push off for Mail, Contacts and Calendars. You can set the default action to "Fetch" and then pick between 30 minutes, an hour, or manually. The iPhone will then only check that often for mail or updates to your contacts/calendars (unless you open their respective apps).
I don't use the Facebook Messenger app so I'm not familiar with its settings, but if you have Background App Refresh off, you will only receive Push notifications and the messages will only actually come through once you open the app. There may be an option within the app itself to restrict the fetch interval for messages (although I doubt it because "Facebook knows best").
Disable 3G for specific apps. If there are apps you only need an internet connection for when you are at home or in a WiFi area, disable cellular data for them. This makes sure they don't sneak any information past you if you inadvertently open them while out and about.
Disable Location Services. This one is always a controversial one. While it will boost battery life and reduce data usage, you have to weigh up the cost of using a little bit of data to be able to use Find My iPhone against the cost of replacing your phone if it gets stolen.


Thats about all I can think of for prepping an iPhone for prepaid use. I'll update this if I think of any more.