E
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:
Missing documents: passport (Inspect mode), entry ticket, entry permit...
Wanted criminal list and photo matchup
Expiration dates
Passport number matchups
Name matchups
Issuing city validity and stamps (1)
Spoken reason vs entry permit purpose and duration (2)
Sex, description
Height and weight (3)
Notes:
(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