I have a Samsung Galaxy S5 and my app menu button is broken. (I accidentally destroyed it while replacing the screen.)

Is there any other way to access the app menu or view a list of running apps, switch between them, close unneeded running applications?

BUILD.PROP SOFTKEYS (ROOT)

If you have root, you can emulate soft keys, like the ones on the Nexus and other devices, by modifying the build.prop file. To do this you will need root. There are many guides to rooting your device, such as this one on XDA.

After you are rooted, install an app like Build Prop Editor. Change (or add) the following key: qemu.hw.mainkeys=0 (add a key named qemu.hw.mainkeys and set the value to 0.) Be careful and make sure that you have typed it exactly as shown. After this, you will need to hard reboot your phone. Power off your device (or remove the battery, not recommended) and then turn it back on. Note: For build.prop edits, you will need a hard reboot; a soft reboot will not apply the changes to the build.prop file.

THIRD PARTY APP

If you do not wish to root, you can install an app like Handy Soft Keys. Please note that this may affect performance, battery, and data usage negatively, as it is not native to Android like the build.prop edit.