If you've done the Lollipop update (I had a feeling you might have) a factory reset is pretty much essential and should sort out all your issues. What happens is that bits of the old system are hanging round and confusing the device. I agree it's scary first time round and it's a bit of a pain, which is why I mentioned the network issue as a possible first step.
Basically, what happens is that you revert your device to its out-of-the-box stage, though still with the updated operating system.
Here are the steps:
1. Make sure your contacts and calendar (if you use it) are fully synchronised. Assuming you're using your Google account for this, log into it on a computer and have a look at Gmail - drop down "Gmail" above the Compose button on the left and select Contacts. Are they all there and up to date?
To check your calendar, click the block of 9 dots at the top rh of the Google screen and then the Calendar icon. Make sure your most recent entry is there. If that all checks out, you're synchronised.
2. If you have photos on the phone, copy them somewhere else, such as a pc. You can do this by connecting the phone via cable and managing things from the pc end. If you're using Windows, when it asks you how yo want to connect, choose the "browse files" option and you can access the phone like a pen drive. Photos will normally be in a folder either called Pictures or DCIM. Just copy and paste them. If you have your phone set to save photos to an external sd card, remove it before resetting to avoid the process erasing the card.
3. Do the same if you have music files or documents. Make a note of your homescreen layouts or take screenshots (which also need to be backed up).
4. If you want to preserve your text messages or call logs, you'll need an app to do this (I think there's one called SMSBackup that's been reported to be good). If you're not worried about these things, skip this stage.
5. Plug your phone into its charger, take a dep breath, go into Settings and find the Reset option (it'll either be under Security or Storage).
6. The device will reset. Various things will appear on screen and there may be some pauses during which it appears to have stuck. Be brave and leave it to get on with it. You have it plugged in so that there's no danger of it running our of battery.
7. When the device restarts, it may prompt you for your Google account. Log in and make sure the "restore device" box is ticked. Your apps should re-install and the settings that are backup up to Google will be applied. You'll have to set your homescreens up again, which is why you have a note of their layout.
I don't think I've forgotten anything major and you should find that your phone is as sprightly as it was when you first had it, maybe more so as Lollipop is generally more efficient.
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