That's the problem I had. For some reason, several taps were needed to edit, click or do an action. Seemed like the iPad was stuck doing something else in the background before responding to the screen input. As for Facebook crashing when loading a picture, this points to a memory issue as there is not enough memory space left to load something else, thus the app crashing.
Make sure that only needed apps have access to the notifications and gps services. I do the same on my iPhone and having all of the apps enabled on these does drain the battery faster and uses more resources. Keeping only those that really need access does help a lot. For example: games do not need gps at all in general, but some do ask for it anyway.
As a test you can start with only one app opened to check if you have something running in the background. Skype, for example, keeps running in the background. If you have several apps like this, you will have issues on the iPad 1.
As you, I did tried all tips and tricks. But what really fixed my issue was resetting and not restoring the device.