I Do not see anything in the settings that I think would prevent the trash can from showing Other than for how long they are to be kept. Settings>Messages>Keep Messages. I have Forever selected. If you do have old messages showing, and obviously you do, I don't see how that setting would prevent the trash can from appearing.
Have you forced closed the Messages app and reset your iPad? Reset all settings?
In order to close apps in iOS 7 or 8, you have to drag the app up from the multitasking display. Double tap the home button and you will see apps lined up going left to right across the screen. Swipe to get to the app that you want to close and then swipe "up" on the app preview thumbnail to close it.
Reset the iPad by holding down on the sleep and home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider if it appears on the screen - let go of the buttons. Let the iPad start up.
You can also try resetting all settings. You will not lose any data, but most of the device settings will have to be entered in the settings app again. Settings>General>Reset>Reset all Settings.