The blue circled item is a jumper plug to fit onto the two pins - usually 0.1" spacing. They are often used to enable or disable some configuration option on installation. They are common on computer boards etc. "J1" stands for jumper 1.
The red circled TAMP component is a tamper detection switch to alarm if the unit is opened. I'd guess the front cover holds it in the depressed position. You've triggered it! Run!
Any idea of how to make the LED on when it's supposed to?
I imagine that on power-up the LED should blink when motion is detected.
When I first opened the cap was in one of the pins instead of both of them. Is that allowed?
The jumper is usually turned sideways on one pin rather than discarded so that it's there if needed later. The jumper may be to switch between two sensitivities or maybe two modes of operation - latch the alarm on if motion is detected or turn it on only during motion, for example.
Also, if I press the tampering again it should work?
I would expect that the tamper switch is in series with the normally-closed relay contact. If either of them goes open circuit the alarm circuit is interrupted and the master panel should give an appropriate response.