Customize Lock Screen

Handoff is why the notes app shows up on my lock screen.

You could try disabling that entirely or be sure that no other device is communicating with your device over bluetooth or WiFi.

If you look at the text in the handoff settings, suggested apps can also be shown based on usage patterns. If you open notes every night at 7 pm, eventually the OS will suggest notes at that time based on inference. Since this intelligence was introduced with iOS nine, it would take a while for the OS to learn your habits so having it appear coincidental with the 9.0.2 update seems plausible.

Everything you've described is exactly how it works in iOS 10, with one exception (noted in the third paragraph).

Pressing the home button once goes to the lock screen. Note that if you're using a 6s or later, the Touch ID sensor is so sensitive that pressing the home button with a registered Touch ID finger generally leads to the device going directly to the home screen. This has been the case since the 6s (and iOS 9) came out.

The new Raise to Wake feature in iOS 10 allows you to raise the device towards you to turn on the lock screen without pressing any buttons. I've noticed that it will turn the screen back off rather quickly if you don't start interacting with it nearly immediately.

Resting your finger on the home button unlocks the phone. The only change is that instead of going to the home screen, unlocking now stays on the lock screen but provides additional functionality since it's unlocked. You can override it staying on the lock screen by toggling Settings > General > Accessibility > Home Button > Rest for Unlock.

Pressing the home button twice does take you to the passcode screen where you can use Touch ID or enter your passcode. Again, if you use a Touch ID registered finger, it will likely unlock the device by accident, and this has been the case since the iPhone 6s debuted.