Thanks for giving some goals. It's really only if you give goals that people can help you.
I'm pretty sure you can achieve all those goals by doing as many as you can in the given time limit, then just upping that number. Actually, some of those look very challenging... 20 pullups in 30 seconds? I'm not sure how helpful those goals may be to your overall goal, though. Going by a previous post, you want to gain enough strength to practice parkour safely and comfortably, right?
If so, all I can say is that some progress is possible without using extra weight, but not much. It's virtually impossible to strengthen your legs without extra weight. Upper body is a little bit easier since there are many more ways to decrease leverage, like one-armed pushups, etc. But in the end you're really going to need to find a weight room somehow. You can also go out in nature and deadlift big rocks, I guess. Or squat with them. Weight room is much more precise and convenient though.
As for rest days, you should have at least one, but no more than two or three ideally, between each workout. As for resting in between sets, I'm not sure. I can tell you that when I'm weight training I often rest 5 minutes in between my heaviest sets. Bodyweight workouts are less intense, though, and so I'm not sure if that same amount of time would be good for you.
Random questions:
Rest days let your body recover. Most of the time, one or two days is enough. In special cases, such as when you are doing very heavy deadlifts, longer periods of rest are required. But you're not close to that stage yet, from what I can gather.
Like I said above, it's very hard to gain strength without weight training. I would go so far as to say every practitioner of parkour, and probably most every other activity as well, could benefit greatly from weight training. Barbells are the favored method of moving weight around here. Dumbbells also work, but are less convenient when you want to add weight. Don't use machines to weight train.
You seem pretty confused about where to start. You've set up a basic routine there, but it does very little to strengthen legs. You are going to need to find a way to weight train, buddy. Oh, one other thing... I saw you asked what core strength is. To the best of my understanding, your "core" are the muscles around your waist- abs, obliques, glutes, hamstrings to a degree, adductors, abductors.
