Your feet should ALWAYS stay in contact with the pedals; whether you use heel up or heel down, it doesn't matter. For general playing techniques, the "ball" of your foot should always remain on the pedal board. If you don't, the beater(s) just goes crazy; and how do you stop or play dynamics? There is also a lot of extra noise from allowing your foot to come off the pedals including the Hi-Hat pedal! There are lots of players who use heel up only; some play flat footed (really fast!). Both work and are valid. I use both, depending on what I'm wanting to play.
I think that playing flat footed gives the most control, yet requires the most control. It takes practice! For me, I practice flat footed a lot, but when I need more speed (more than I can give using "flat feet"), I lift my heels (only an inch or so). This allows me to play by pivoting my ankle ever so slightly and using my toes (ball of foot). Sometimes I even slide my foot back and forth or side to side, being careful to always remain on the pedal board. Now it is just second nature.
I view foot technique just like hand technique!
In hand technique you use either arms, wrists or fingers and any combination of the three; all depending on what you are trying to do. If I need to play soft and fast, I use more fingers. If I need to play really loud, I use more arm to pick the sticks up higher; not to necessarily hit harder!
The same with foot technique. You use either flat foot, heel up with ankle pivot or heel up and leg thrust. If I need to play faster than what I can do heel down, I use heel up and ankle pivot. If I need to play fast with a lot of volume, I use heel up and the weight of my leg (working with gravity) to thrust the beater.
I teach my students to always allow the beater to bounce off of the kick drum / bass drum head. This seems to produce a fuller sound plus allows the beater to come back to a resting position, ready for another blow. Think of it this way: if you were in a fight, and you were planning to hit some guy square in the kisser, would you punch and push your fist two inches into his face, or would you punch and naturally allow your hand to bounce off so that you are ready for another punch (just in case you were going to have to hit him again and again)? The correct answer, I believe, is to allow rebound! The same is with sticks, when you allow rebound the sound is much fuller and pleasing; rather than slamming the stick into the head and muffling the sound. Don't get me wrong, sometimes you need the sound that is produced by ramming the beater into the head and leaving it there; it all depends on the sound you need. I want to have control over my feet to do what I want, not to do it because of habit or because I have no control. One of the real benefits of "pushing" the beater into the drumhead is that you get a consistent, compressed, punchy sound. Develop both techniques and be able to distinguish the pros and cons for both.
In regards to spring tension, here is my opinion. Every person is different; we are all built differently and weigh differently. I teach my students to set the beater/spring adjustment so that when the pedal is just sitting there (your foot is no where near it!) the beater COULD be back as far as being parallel with the floor; depending on how loud you want and need to play. This part is crucial: WHEN YOU PLACE YOUR FOOT ON THE PEDAL (relaxed, just lay it on the pedal), THE BEATER SHOULD ONLY GO HALF WAY (the distance from the beater all the way back, to all the way forward, striking the head) TOWARDS THE DRUM. If you place your foot on the pedal (don't push), and it is touching or only a inch (or so) from the drumhead, you need to tighten the spring and/or re-adjust. If you place your foot on the pedal (relaxed, don't push) and it barely moves (you would have to really push to move it) the tension is too tight. The older I get I realize that the beater needs to be able to "swing freely" as much as possible, but not too much. In this way, you will be more relaxed, not burn so many calories, and be able to play longer without tiring!
The above is only my view; however, it does work, not only for myself and my students, but for many professional players. Watch and listen to several pros who do what you want to do. Copy them and apply it to your playing. Above all, stay relaxed! The "big boys" make it look very easy! Why? Because they are relaxed!!!