Any wire connector can fail. Screws can loosen with vibration, stress, corrosion, temperature fluctuations, and so on. Push-in (or stab-in) connections can fail for the same reasons. Crimp and pin connections, friction lock, and even soldered joints can fail.
The real question is: what's the most effective wire connection given the operating conditions, parts and installation cost, lifetime expectancy, use case, and so on.
For example, where I work (we manufacture large electric signs), we use all manner of connection methods, and they vary based on these factors. One such factor is serviceability. If a connection is expected to be rugged and reliable but disconnected for safety reasons, we may use screw terminal blocks (which only accept straight wire) or friction lock (which requires use of a screwdriver to release, similar to the push-in connectors on outlets and switches).
Outlets shouldn't require frequent service nor replacement, nor should experience extreme vibration. They also won't (usually) have wide temperature swings, which leaves the stress from normal usage (connecting and disconnecting plugs) which can wiggle things loose over time.
A well-made outlet can have quite good push-in connectors; by "good" I mean they hold solidly for years and aren't likely to fail with normal use. However it can be difficult to tell during installation whether the inserted wire has made good contact or not. Also if one happens to be faulty, it may accept the wire but provide little feedback that the locking mechanism is less than optimal. Pulling on the wire after installation is one way to tell, but if you're using the push-in connectors, are you spending the extra time to do the pull test? Probably not. Screw terminals are easy to visually inspect and determine if solid contact is being made and that the wire is firmly held by the screw or plate.
So while push-in or friction lock connectors can be quick and useful in certain applications (with appropriate confirmation of operation), I recommend wrapping wire around the screw terminals for the most durable connection that's easily verifiable. Usage and vibration can still loosen the screw, and in such case the wire hook will be less likely to separate from the outlet than a straight wire. In all cases where the screw comes loose, however, arc fault can occur, causing problems.
That is tar from the old insulation, that's why it is sticky. I don't know why or how it happens, but old insulation had tar in the paper, and it is very common.
It is not scorching or melting from overheating.
Take a utility knife and try to gently scrape some of the black away. You should see it's just on the surface. It's nothing to worry about.
Best Answer
The first (2) photo's show a Female "Bullet" crimp connector. The third shows a Female "Spade" connector. Both are available on Amazon.
They are sized according to wire gauge; that is the color coded crimp connector's only accommodate certain wire diameter's.
You may want to consider that a new bullet connector may not fit securely enough to stay in place. You may want to consider soldering the wire at the break and wrapping it with shrink tube. I've made repairs similar to your's and had to reduce the opening of the new bullet to keep it secure.