This is one of the biggest grinds in the game, but there's no specific guide in order to do all of these, as Gauntlet mode is pretty much random, but here are some tips for some of the bigger pains of it.
I highly recommend stacking a lot of the challenges that involve killing X officer, X amount of enemies, X amount of officers, etc. This will greatly reduce the amount of time spent on it.
As far as your team goes, I used Xu Shu with a 5 Star weapon, specifically his 5 Star that you create through Gauntlet mode, with Brawn, Windslay, Flak, Courage, Agility, etc. The other characters I used were Sima Zhou, Lu Bu, Orochi X and whoever I wanted to level up. The reason I recommend Xu Shu, is that when I promoted him a few times, pumped up his speed and put on items like Charm of Alacrity and another speed booster, not only did he have 999 Attack, but also had 250+ speed, making him faster on foot than most mounts. Not only that, but put some items into Musou recovery, and he'll be unstoppable. His R1 attack basically breaks the game, making it manageable at lvl 120 Miasma.
For some requests, like kill 999 Enemy Officers, the first DLC Stage (accessed after completing Gauntlet mode once and by pressing X on your controller) is great, as there are TONS of officers spawning like crazy.
For the Repel 500 of whoever, you can alter the levels by using 5 keystones, but I don't recommend this until you've exhausted your options. Some enemies, like Escape Ninjas, only appear on Purple Keystone stages, while others like Glacier Swordsmen and the like appear on Yellow.
For enemies like Archers, Hurlers, Tigers and Wolves, they don't appear on the mini map as a group of enemies like Rikishi. In these situations, it's best to run around until you spawn a group of them from a Miasma Pool to maximize efficiency instead of running around willy nilly looking for them.
The biggest, most painful grind, was 999 Famous Officers and 999 Otherworld Invaders. For these, stack them as all Otherworld Invaders are Famous Officers, and they count towards both. The best levels I would play for these are the levels where the name is glowing in Red Text. On these levels, they spawn at a higher frequency (a group of 5 every 5 minutes). The best thing I would do is choose a large level. On large levels, you'll often see "A treasure chest has been changed by the surrounding miasma". When this happens, the chest will change to red on the mini map, and when you open it, it will spawn 5 Otherworld Invaders. On larger levels, like Toushi, I would often get about 40-60 invaders in about 20 minutes. Even at that rate, it'll take HOURS to complete all of the requests. On the levels where you're grinding these invaders, find a garrison with a climbable tower. Enemies won't be able to hit you or climb up. Just sit there once you map out the level and wait for the chests to change or the invaders to spawn, then run down, wreck them, and run back.
Good luck achievement hunters!