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Judging from what you describe,

My guts feelings is these cases

Patellar tendonitis and Torn Meniscus

Pattelar tendonitis is what we called jumper knee. From over training, the tendon get inflammation and you can't bend your knee properly. The pain actually feels like under your lower part knee cap.

The other one is your meniscus kinda like cushion which sit between your thigh bone and shinbone got torn and not as smooth. Pain, click and popping sound will occur when your bend your legs.

I personally got "jumper" knees, and the only remedy for me is to rest and rehab my legs for three months.  Torn meniscus can heal over time, however some case required simple surgery to fix the problem.  My advice will be seek a doctor. Sorry.

It happened to me, and happens every now and then... not to the point of falling down but you get the idea - my quad sometimes forgets to fire and lets the knee collapse forward/down... it happens. It's usually when I feel lazy, and I kind of feel lazy all the time.

If you're tired chances increase of this happening.

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2)  Is it really impossible to gain muscle mass while losing some body fat?  If not, how do you do it?

Of course it is, a very simple way is keeping protein extremely high, fats to a moderate to high level (depending on activity levels), and controlling carbohydrate intake.  Moderate the volume on your lifting program so that you are still getting stronger, sometimes this requires a lower frequency of lifting and sometimes you dont have to change much, but keep a PREMIUM on gains in strength in the weight room.  You will be surprised how much protein and fats you can eat if you limit the carbs to a bare minimum and still build lean tissue while losing bodyfat. 

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4)  I've constantly been hearing conflicting information about core/mid-section strength in relation to athleticism/vertical jump.  Can strength training your abs/obliques/lower back really increase your athleticism/vertical?

Not to the degree that its hyped by a lot of "trainers", and even then its an indirect correlation.  The core strength can help you maintain good athletic positions on the field/court and in the weight room by enabling better pelvic control and core stability, which will help, but doing med ball twists and planks wont give you 5 inches on your jumps by itself.

Well, to be a better athlete you don't have to have a really low body fat %.  However, if your looking at losing bf not weight, I'd say focus more on high intensity rather than standard "jogging" type cardio.  Here are a couple of suggestions:

1. Swimming - 10 x 50 m sprints Freestyle (and similar swimming sprint focused workouts)
2. Boxing - Heavy bag, speed bag, shadow boxing, etc
3. METCON (Metabolic Conditioning): Crossfit or gymnastics type workouts, for example: Run 1 mi, 100 pull ups, 200 PUSHUPS, 300 Squats, Run 1 mi for total time

If you add two of these type workouts per week to your current plan, you'll be in great total body shape.  Tough to recover so be careful.

i did look at that, from what i could make out it focusses on restoring mobility + strength...it says "The H.E.M. ankle rehab system will get you out of pain and walking normally again in about 3 - 7 days (instead of the typical healing time of 4 - 8 weeks with just rest and ice)!".....see JUST REST AND ICE....rest and ice is not rehab....that is the first stage but i believe..depending on the significance of the injury you can begin mobility work...drawing alphabet with toes/circles/plantar flexing back and forth and then move onto strengthening as soon as pain allows....

i saw a very good(well i think so anyway) sports physio when i had my injury and he had me doing mobility and band strengthening work before i could even walk without the use of  a walking boot....which i think drastically improved my recovery time....because by the time i was out of the walking boot i had the strength to walk properly (after a few weeks in a cast my calf had basically disappeared).

im by no means any expert but ankle rehab depends on much more than rest and ice - although you should still do these

my rehab consisted of

complete immoblisation - 2 or 3 weeks....
in the 4th week i began mobility (plantar and dorsiflexion) and strengthening (plantar flexing against resistance band VERY high volume..literallly sitting there for hours doing it while watching tv or such)
the next week i moved onto inversion/eversion mobility and band work....and did this up until i got out my walking boot after 6 weeks.
then i began with calf raises and wobble board exercises while carrying on with all the band strengthening
in maybe the 8th week i was moving onto hopping exercises 1234s around a cross etc.....i began light jogging around this time too
after 10 or 12 weeks i was back playing...which i thought was remarkable...noone believed i would be playing basketball just 12 weeks after that injury. another thing that helped alot was active mobilisation from the pysio....physically forcing my ankle rom back

obviously for a sprain the rehab process can be accelerated alot...but its important not to rest completely...rest from dynamic activities..hopping etc until you can do them mostly pain free but mobility/strengthening stuff can begin very shortly after the injury in most cases i think

when i have more time after exams and stuff ill put together a more easy to read detailed post...ill be doing my work experience for 2weeks with the physio i saw soon too so ill be nagging him all the time for info! sorry for the long drag of a post that is probably unreadable