I know you guys are probably getting sick of these, but for the past few months I have been stuck in a rut workout-wise, and I feel like I need something new to get me going again. So I am asking for you guys to help me out. Here are some goals I want accomplished:
Do 25 pushups (I tried right now, I can do about 6 properly

) <-- Will this lead to a gain in benching, or is that seperate?
Do 5 dips (Not even 1 right now)
10 Pullups (I can do 1/2 a pullup, almost 2 chinups)
Run a 5K (that's 3.1 miles, right?)(I can run 1.5 mi in about 16-18 mins right now)
Decrease mile time to sub 8 mins (right now its at 11 mins)
Increase core strength (I dunno specific goals for this, so...)
Increase back strength (Rowing is good I've heard, but I don't know how to go about this.)
Increase agility and speed.
MAIN GOAL: Burn some fat off of me, gain some muscle, get healthier. Need to get fit for parkour

I know, it seems like quite a lot. Well, it is

. Another thing, very important: I am constantly stressed for time, because I am in the IB program at school, which means I get about, if not more than, 4 hours of homework every night. So I was thinking of incorporating tabata intervals, but I don't know how to go about that. GTG would kind of be unrealistic, since 7 hours a day I am at school. But I suppose I can work a way around that (pushups in between classes lol). And I want to make sure that this will provide significant strength gains (which I'm pretty sure it will.).
Question: Would this type of bodyweight-esque program provide more muscle gain than a dumbbell/barbell program?
thank you verrry much.