How does the oscilloscope trigger really work?
When the signal measured then hits 5V, the scope's ADC activates and it starts to sample the signal. Some amount of data points are gathered, and these are plotted on the screen.


The scope's ADC is continuously running and gathering data. The trigger controls what is displayed.


  Then there is a small "dead time" after which the scope again waits for the trigger condition to be met, and the same amount of data points are gathered again. These should now line up with the previous set of samples, and therefore the scope output looks stable on the screen.


This is only the case if your signal is perfectly periodic, and your explicitly only displaying triggered data (many scopes have an "auto" trigger feature that will display data even if the scope hasn't been triggered). As mentioned by Hearth in the comments to my answer, the "dead time" you describe is called holdoff, and correctly setting this is essential when triggering on certain waveforms. For example, a periodic signal with two quick pulses followed by a long delay would require a holdoff long enough to ignore the second pulse (so the scope doesn't re-trigger on the second pulse).


  The time axis is something I don't completely understand. I believe that the origin of the grid, where the highlighted dotted lines intersect, is the triggering point. At that point (at "t = 0") the voltage should be equal to the trigger level voltage. Am I correct so far?


Yes.


  The thing is, this is not always the case with my oscilloscope. Sometimes the voltage at the origin is not equal to the trigger level, and the signal even drifts slowly to either direction. What causes the signal to drift even if the trigger is set?


The x-axis is movable on most oscilloscopes. If you look closely at your screenshot, there is a white arrow at the top of the screen pointing down. That is your horizontal (\$t = 0\$) reference. You'll also notice a yellow arrow towards the left pointing right that shows the currently set trigger level.


  Another confusion that I have: I've seen the right side of the origin called the "post-trigger" data and the left side "pre-trigger" data. How is there data from before the trigger, if data gathering starts from the trigger? Shouldn't the trigger point actually be at the very left of the screen?


The scope continuously captures data, but only displays data when the data it captured meets the trigger conditions. Based on your horizontal position, the amount of post-trigger or pre-trigger data displayed will vary.