I can't have the slime follow me and I can't go past the basement code area thing
To get the slimes into the party, you need to have them at the lab, feed them slimeballs until they reach "adult" size.
After that, you need supercharged slime from Joe. (help him out 5 times first to get that area unlocked)
Feed them 3 lots of supercharged slime and they'll size up again. Then you can add them to the party. However, there's an issue with the slime pictures being incorrect, so you'll get a mass amounts of errors while they are in your party (until the next public version ~ roughly 1-2 weeks)
In the castle? The passcode is contained on 3 hints found on the second floor. Or just beat the Security drone that spawns when you get the passcode wrong.