A lot of split routines recommend training chest with triceps and back with biceps. Doesn't fatigue play a role when working out triceps and biceps after a relatively tough chest and back workout respectively? Also, can biceps and triceps be trained before back and chest respectively to alternate the routine and providing equal focus OR should they always be trained afterwards?
Yes, fatigue plays a role when weightlifting, and it should be taken in to consideration when doing any sort of strength training. Muscle building is determined by intensity (the amount of weight carried per rep) and volume (the total combined amount of weight carried). The higher the volume and intensity, the higher the muscle building capacity (up to a point). As muscles fatigue, they are less capable so intensity and volume drop.
However, split routines like the one you described are generally not that hard, so you own't fatigue as much. Especially as you adapt. The reason you would do chest and back first is because the exercises that focus on these are generally more "full body" meaning that... yes while it does focus on your back, you are also working your forearms, biceps, and triceps. Think of a standing barbel row, for example. It's impossible to perform it with just your lats. You have to hold it and row it which uses your arms. Curls on the other hand are just bicep exercises with some small shoulder and forearm usage. Fatigue is less of an issue.
It's recommended to do full body exercises first before anything else. E.g. Squats, deadlifts, bench presses, standing overhead presses, barbell rows, etc. They take more energy to perform safely, and are more effective when you do them with full force. Only after you do the hard stuff do you do the easy spot exercises. In that case, only volume matters so you can lower the intensity which is required because of the muscle fatigue caused by the full body work.