Since running out of battery on my Mac, fonts are missing and apps don't open properly This first happened when I ran out of battery on my Mac. When I charged it, it was acting very weirdly. My desktop background is black and when I login the background is black. When I open Hamachi it says "Failed to connect to engine", Stuff is missing such as desktop pictures and letters that I used to see are now a box with a question mark "[?]".
I would try resetting your SMC

System Management Controller (SMC), a circuit that deals with power management, temperature monitoring and fan control, status lights, keyboard backlights, and a few other components. If your SMC becomes confused, you could experience problems like excessive fan noise, slow performance even though Activity Monitor doesn’t show the CPU being overtaxed, apps that take forever to launch, batteries that don’t charge correctly, problems with sleep or wake, and so on.



Before you can reset your SMC, you must shut down your Mac. After that, the procedure varies depending on the type of Mac you have.


Desktop Macs: Disconnect the power cord (either from the Mac or from
the AC outlet). Wait 15 seconds and plug it back in. Then wait
another 5 seconds and turn the Mac back on.
Portable Macs with non-removable batteries: Make sure the Mac is
plugged in to AC power. On the built-in keyboard, press and hold the
Shift, Option, and Control keys on the left side and press the power
button. Release all the keys at once, and then turn the Mac on
normally.
Portable Macs with removable batteries: Disconnect the AC power cord
and remove the battery. Press the power button, hold it for 5
seconds, and then let go. Put the battery back in, reconnect the
power cord, and turn the Mac on normally.


You can also try starting up in safe mode

Follow these steps to start up into safe mode.


Start or restart your Mac.
Immediately after you hear the startup sound, press and hold the
Shift key.
Release the Shift key when you see the Apple logo appear on the
screen.


After the Apple logo appears, it might take longer than usual to reach the login screen or your desktop. This is because your Mac performs a directory check of your startup disk as part of safe mode.

To leave safe mode, restart your computer without pressing any keys during startup.

If you don't hear a startup chime after pressing the power key, see what to do if your Mac won't turn on.

If you're using FileVault
If your startup disk is encrypted with FileVault, you can still hold down the Shift key immediately after powering on your Mac to start up in safe mode. You might be prompted to log in twice as part of this process – first to unlock the startup disk, and a second time to log into the Finder. You can let go of the Shift key after you see the first login screen.

If an issue doesn't happen in safe mode

If an issue doesn't happen when your Mac is started in safe mode, try restarting again without pressing any keys at startup. If the issue appears to be resolved when you start up normally, it was possibly caused by a cache or a directory issue with your startup disk that safe mode fixed.

If you restart your Mac normally and an issue comes back when you reach your desktop, try disabling any login items that automatically open when you log in.
If your Mac restarts or shuts down in safe mode

If your Mac automatically restarts in safe mode, OS X might have found an issue that it's trying to fix. 

Safe mode performs a directory check of your startup disk, similar to what happens when you choose to verify or repair a disk using Disk Utility. If OS X finds an issue, the directory on your startup disk is repaired and your Mac restarts. 
If your Mac turns itself off when you start up in safe mode, check the power connection on your Mac to make sure it's plugged in securely at both your Mac and the power outlet. If you're using a MagSafe power adapter, make sure the LED on your power adapter is yellow or green. Then, try starting up in safe mode again.