One of the first things that I try to notice is gender, then necessary documents, then picture, then issuing city, then expiration date. I don't think I've seen the second half of the game, so I also wonder if new conditions could change that order. I know a condition is added that involves checking faces even before checking the face in the passport, so an answer would note checking this before the passport but also noting that it isn't necessary at first until what condition is added.
Yep! You've basically got it. There isn't a strict best order "Firstly ..., secondly..." because every person is different, but you've got your bases covered.
You want to streamline your work by looking for the most common fakeries first. As a definitive statistic breakdown on errors doesn't exist, I've ordered the list according to my opinion in the most common discrepancies:
(1) Late game, having a color cheat-sheet from the wiki is pretty imperative for doing well. A cheat-sheet simply lists each country, their cities, icons, and seals.
(2) (if you have a short auditory memory, consider bumping this up if the audio transcript takes too much time for you)
(3) I personally have a hard time with height and weight and rarely check them for streamline purposes as they are perhaps the rarest discrepancy
I suggest you start with checking the possible discrepancies that you check the fastest. Personally, checking issuing city takes a long time for me, so I check that one last. If there is some other discrepancy that I see before I come to the end of my list, I don't have to waste time on the slow issuing city check.