Concerning the black cover, there's a machine in the dungeon usually on the same level as the pit that gave you the black cover that will only give an option to someone that is covered in the black goo. it says it will remove it. As to the Divine set it's basically luck as to whether or not you get the real one or as Secretsakura stated, the ones that come from dungeon completion rewards are always what they say they are so just keep an eye out for it as a reward.
Razule
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This is just a guess as I haven't brought one to it myself but I found in another area a gate asking for the "lowest" parasite. Your gate I'm guessing actually asked for the "highest". Now like you I had someone that had the full parasite set. However, I didn't have anyone with me that in the actual parasite slot(I did have the set version) had a "heel parasite" which I'm guessing is the lowest and the brain parasite of course would be the highest. My guess is you need to have someone with that parasite in their actual parasite slot. That being said if you run around enough and check each hole you can drop down you will eventually find the one that leads to the boss room without ever going through those doors.
This is correct. you do not need an empty slot. Also, as a tip if you don't want to wait until the dungeon is lower level, just know that it will remain there even if you fail to complete the dungeon (had to retreat) and you can try again. Also rescuing them and then fleeing the dungeon still grants you the rescue and closes the dungeon, you just don't get the item the dungeon was offering.
Important to note that you can't upgrade the rats. whatever level the cursed ring is that you get to force your character into a rat/slim/goblin/whatever it will always be that level and no further skills or progression on them will upgrade the class nor give you more fighting options. Which I find really sad actually.
I haven't even used alchemists except where required by quests. I normally run a warrior or slave up front, slave/warrior/melee pet/tank-melee cow/paladin 2nd, healer or ranger/throwing dagger rogue/barking pet or buffing maid/bunny in 3rd, same options for 3rd in 4th. I guess it just depends on what you find.