1. However you're able to get the fruitcake to cure your Sim, there's one thing you should know. They have to eat ALL of it! Every single serving in one sitting in order to be cured. Having just one slice won't do it. They won't like it, they'll complain and be miserable doing it, but that's the only way to fully cure them.
2. One of my current Sims is in the Doctor career, however, it's only the first time I've ever played a Doctor. I did do some research about it before having her start the career though and found out a few things. What someone else said is correct, having a higher level in Logic definitely does help with diagnosing illnesses. Plus, every illness has a set of symptoms that makes some of them easier to cure if you watch that Sim for the signs. Here's a list of the illnesses and their symptoms. Hope it helps your Sim, it's helped immensely with mine. I found this list on SimsVIP, I think. lol I checked out so many websites it's hard to remember. Anyway...
Sweaty Shivers - Itch, No Rash, they might fan themselves as if hot and thought bubble shows a thermometer
Bloaty Head - Headache (headaches look like a red pulsing fog around their head and Sims might press their forehead), Steam coming out of their ears, maybe a rash
Starry Eyes - Dizzy, stars above head, swatting, maybe a rash (*Note - swatting at the air as if there's an annoying insect around them is exclusive to this illness. They might not have the stars above their head either with the dizziness, but if they're swatting at the air, it's this illness)
Llama Flu - Cough, Sneeze, gas cloud, maybe a rash. Watch for a thought bubble showing a box of tissue, plus it's the only one the Sim will sneeze, I think.
Gas and Giggles - Fart and Giggle, maybe a tiger-stripe rash. (*Note - sometimes ill Sims will laugh when they're nervous, they should laugh a lot like an erratic Sim for this illness. I have never had one with this illness actually fart either. Maybe I just haven't noticed though. lol)
Triple Threat - Dizziness, coughing. Lack of swatting & lack of fever. Requires surgery. (Haven't had this one so far)
Itchy Plumbob - Itching a lot, rash (*Note - not itching just a little, itching a lot and all over themselves)
Burnin' Belly - Visible stomach pain, maybe fever. Requires surgery. Sims will hold their stomach and moan with this illness. (Haven't had this one so far)
*Note - You probably already know this, but make sure you know what the illness is before trying to cure them. If you're wrong, your Sim will take a HUGE performance hit. What I've been doing if I'm not sure or if all the tests are inconclusive is to just move on to another patient. I try to pick patients showing clear symptoms for the above illnesses to make it easier on my Doctor, well she's an Assistant Nurse right now too. :)
3. I don't have this problem in my game very often and don't use any Mods, so I can't help with this one.
Happy Simming! <3
EDIT: Oh! Another thing that helps with diagnosing patients is to be in a focused mood. I had my Sim paint a few Mathematical Diagrams to put on the wall in the patient area and the rooms with the x-ray machine/treadmill, plus the room with the analyzing machine and computer. Have your Sim Ponder Moves on a chess table too before leaving for work. If you have access to TS3 reward lamps, could always just put a few of those in those areas instead. OR get some elements to put around the hospital. Those have a focusing aura as well that you can enable.
