When reassembling the logic board into the case, watch out for the tiny nose of the audio jack that's facing towards the center of the device. This nose has to be inserted first, then the rest of the audio jack followed by the logic board itself.
This step is crucial to getting the proximity sensor back into working state:
There are two sensors on the power and sensor cable peeping through the front glass of the phone. Looking at the front of the phone, the sensor on the right is a square and dark one, whereas the sensor on the left is a round golden one. The square, dark sensor on the right needs to be covered with the proximity sensor UV filter foam sticker everybody is talking about. The sticky side needs to be applied directly to the sensor. You can get these stickers at eBay. Don't apply the sticker to the front display glass of your phone. Don't apply the sticker to the golden, round sensor.
Before beginning the unsoldering I cut all battery wires near the battery to prevent shortening battery or board. I then commenced to step 17, 18 and 19 with the battery already detached. Instead of pulling the wires out of their soldering holes with the battery I used a little plier to pull each of them out. Holding the soldering iron to the pad on the other side while pulling with the plier makes that quite easy.