Ridiculous number of unnecessary empty birthday calendars that can't be deleted
As Insomniac Software mentioned, the Birthday calendars come from the Contacts app.

In the Contacts app, it (confusingly) launches at a page that shows an aggregated view of contacts from different sources.  Tapping the "Groups" button at the top left of the screen takes you to the list of sources.

You mentioned that on your Groups screen, you have two sources of contacts:


Facebook.  Birthdays in this group show up in the Calendars app in the "Facebook" section, not in Other, so that's not the cause of the issue
On My iPhone.  Birthdays in "Other" come from here.


The problem is that, from what you've said, it sounds like your contacts are stored on your iPhone, not in iCloud - otherwise they would show up in an "ICLOUD" group, not an "On My iPhone" group.

To solve your calendar issue and to prevent potentially losing your contacts:


Select a place to store your contacts.  I'll give an example for iCloud because I suspect your contacts were there and got dumped on your phone while changing settings.  If you think you have them stored in gmail, post a comment and I'll update my answer.
Move the contacts to your selected location.  To merge them to iCloud: go to Settings > Mail, Contacts, Calendars > iCloud, turn on "Contacts".  It should tell you "Your contacts will be merged with iCloud".  Tap ok to allow it.
Go back to the Contacts app, tap "Groups".  You should see an "ICLOUD" group.  It may take several minutes to sync - go get coffee and come back if you don't see it.  You should not see an "On My iPhone" group.


Now, you need to make sure your contacts are there.  In the Contacts app on the Groups page, to see what's in a particular source, tap the boxes under each group heading until the checkbox is only on the one you want to see.  So, to see what's in your iCloud contacts, there should only be a checkmark on the box under the "ICLOUD" group heading.  Then, tap "Done" in the upper right corner to see the contacts - it'll only show the contacts in the selected source (i.e. iCloud).

Then, open your Calendar app, wait a minute for it to sync, and tap Calendars.  If your 8 Birthday calendars are still there:


Close the app, wait 5 minutes, check again.
If they're still there, continue.  If they're gone, you're done!


If your Birthday calendars are still there:


Go to Settings > Mail, Contacts, Calendars > iCloud, turn off Contacts.  Tap "Delete contacts from my iPhone" (very important).
Wait a minute for it to sync, then go check your Calendar app.  The Birthday calendars should be gone.  If they're not, give it a couple more minutes.  If they're still not, turn your contacts back on anyway (next step) and comment.
Go to Settings > Mail, Contacts, Calendars > iCloud, turn on Contacts to get your contacts back.
Wait a minute, then go check your Calendar app.  The Birthday calendars should still be gone.