Going to messages directly from locked screen

To allow replying without your passcode try:

Settings > Notifications > Messages > have show on lock screen on and show previews set to always. 

Settings > Touch ID and Passcode > Turn on reply with message under allow when locked. 

When a message comes in, swipe it to the left on your lock screen and tap view. You should be able to reply without your passcode. 

If you swipe the message to the right and tap open, you will be prompted for your passcode.

-AJ

You've never been able to, as the messages app requires the phone to be unlocked to open it.

That should not be possible unless your phone is jailbroken. Apple's security model does not permit any app to be opened from the lock screen, unless you don't have a passcode on the phone.