*I'm going to assume you're talking about a normal playthrough and not a no punishment one.
Unless you're talking about purposefully doing a slum only run, you can move into a district after you do a contract for a faction.
Starting in the slum is definitely the hardest, but if you're not hell bent on picking the most difficult slaves, it's not that bad. Even if you do want to do a slum only run it is manageable, but you'll be very dependent on the Salon and bathing as a reward to keep up hygiene. (Also, training witchdoctors and servants are basically School dependent, since you don't have a kitchen or lab.)
On the issue of strength, you simply rest. If you keep yourself in good health and the house tidy without exhausting yourself, your strength improves by itself, no need forcing it in the arena. If you're not overexerting yourself it basically takes 3 days for your strength to become C.
You give gifts according to merit, and if for some reason her spoiling rises, you put her in place, don't reward her when she deserves it or make her sleep on the floor. You don't have to keep her always happy, you just have to avoid her going into a depression spiral.
Finally, rules. You already have 2 rules which she can't even break at the beggining. No masturbation and deny orgasm. You just have to get a 3rd that she obeys and you're basically done. (You don't even need a 3rd if you're playing on Normal.) You can also just force rules if you buy the items for it. Force rules also counts as a rule.