I am a beginner at minecraft. I am very confused by the following setup:
If I step on the pressure plate, piston number 1 is activated. The key piece of redstone is the one circled in blue; if I remove that one, the piston stops activating.
As for piston 2, I am also confused why it's not activating. I thought it should be diagonally powered, but it's not.
It is being powered via quasi-connectivity. An interesting quirk with redstone which is not a bug. It is an interesting one, I will use pictures to demonstrate your circuit but if you click that link to go to the wiki, it is filled with several different ways quasi-connectivity works.
In your case, you have created a Block Update Detector (BUD) that immediately updates the block via the redstone dust you have circled in blue.
Here is an example of your current circuit.
The yellow block powers the piston via quasi-connectivity. At this point the piston would not extend, however, the redstone dust on the purple block immediately updates the piston which causes it to extend. The dust is not powering it, but updating it.
While still powered, removed the redstone dust on purple block:
The piston remains extended as it is still powered via the yellow block.
Now we cycle the lever off and back on:
No extension of the piston because the dust on the purple block is not there to update the piston.
Manually placed an orange block next to the piston: 
The placing of the orange block next to the piston while the circuit was powered caused the piston to update and extend.
Unfortunately you can not have your pressure plate powering that block.
Piston 2 is not extending because you are powering the block lower then the piston.
If you were to raise the level up one, so you are powering the block next to the piston, it will extend.
