Broken connectors / pins on graphics card
What you’re seeing is actually not a broken connector. These pins are intentionally kept short so they connect last. Remember that PCIe is hot-plug capable, you can add (and theoretically also remove) cards in a running system. If you look inside a USB plug, you can see something similar: The data pins are shorter than the power supply pins.
The short pin near the rear of the card is “PRSNT2#”. The one to the front is “PRSNT1#”. It’s connected to the rear pin and informs the motherboard of the length (in PCIe channels) of the card. There are multiple possible “PRSNT2#” locations, one for each possible card length.
You can read more on the PCIe connector on Wikipedia.
Stackcraft_noob is right about bad connectors that can damage your system, however, he is wrong about this case. NOTHING is wrong with these connectors, as Daniel B states, this is how they are manufactered. It's (almost) always the first pin from the right on the back, and the second pin from the right on the front.
As you can see here: 
And here:
Just google any recent graphic card from AMD or Nvidia, and you will see all cards have this. If you see a card that has not, you are looking at a render.
Can broken connectors cause bugs, errors etc.?
Yes! Especially the inside of your computer is very sensetive for broken contacts. Doesn't matter if its one, one-half or ten. It causes that the contact can't anymore send clearly siganal from the motherboard to the graphics card and then you get errors.
A list of possible errors that a loose contact could cause:
- Causing other errors and problems
- Short circuit
- Software errors
- Damaging other hardware on board
- Overclocking
- Overheating
- etc.
As you can see in your example this broken slots/ connetors caused "The card was sometimes not detected in the Slot and stopped working randomly"
So, what can I do?
I don't know but how this happened! But if it isn't caused by you (for eample thought such violent action or carelessness using) then it's a error caused by the producer. So if you still have warrenty on your graphic card, pci/pcie-slot or whatever I would send it to him to fix this damage!
However if you don't have warrenty or if its your falt, then you should send it to the producer or any technical workshop to repair it. But then you need to pay for that. So if the repair price should become too expensive you should think about just to buy an new graphic card ;-)