Repeatedly Prompted for iCloud Password

I have an iPad mini with iOS 7.1.2 installed. It was new out of the box two weeks ago (warranty replacement for one that had a different problem). I didn't restore from any previous backup, but I did connect to my iCloud and iTunes accounts before re-downloading the apps I wanted and syncing with my iTunes library to add photos and other miscellany.

For the last several days I have been prompted two or three times a day to enter my iCloud password when I unlock the device. There is no discernible pattern, but so far I have not been prompted for it while using the device - only immediately upon unlocking. I have tried the following, based on suggestions in other forums: (1) delete and re-connect iCloud account; (2) delete iCloud account, power device off and on, re-connect iCloud account; (3) delete iCloud account; restore network settings; power device off and on; re-connect iCloud account. The problem persists. Today, for the first time, I received a notification while the device was locked alerting me that I needed to verify my iCloud password for iCloud backup.

The device otherwise operates normally, with one, one-time exception that I mention only in case it presents a pattern that I'm not experienced enough to see: yesterday, with cellular data off, I was connected to my home wifi and browsing with no problem. Without warning, the wifi connection dropped, or appeared to drop. I went into settings and verified that wifi was enabled, and that my home wifi network was selected. It was, and the settings panel showed a connection (but there was no wifi symbol in the status bar, and no actual connection). The wifi was working properly; both my phone and my laptop, within five feet of each other and the iPad, were experiencing no connectivity issues. Switching to airplane mode and back did not solve the problem, but powering off and on did, and it has not recurred.

I would appreciate any assistance. I have not yet made a Genius Bar appointment, as there is no guarantee that I could replicate the problem in-store (though I do have screencaps of some but not all of the prompts). Thanks very much.