There are so many varied opinions on what a "good" push-up should consist of, I thought I should throw in my philosophy as well.
Speed:
Fast or slow, doesn't really matter. Well, it DOES matter... but you can't say that either is right or wrong. I personally prefer fast push-ups, but that is just me. With any strength exercise, I go for fast and explosive as opposed to slow. This is commonly referred to as plyometrics, but I'm pretty sure that term was just coined to sell workout videos. If you are in a "most reps" competition, then it certainly makes sense to go fast. (assuming you don't sacrifice technique for speed) It really all comes down to what is called "time under load". How much time are your muscles actually working during the workout? If going slowly, you may get around 12-15 reps in one minute. While the rep count is low, you are still getting a decent workout because of the time under load. On the opposite end of the spectrum, it is not extremely difficult to do 15 reps in under 10 seconds. While your time under load is considerably less than had you gone slow, you will still get a much greater workout than going slow for only 10 seconds. I try to do some slow reps and some fast ones. There is no definitive answer here, and anyone who tells you it is "correct" to do it one way or another doesn't know what they are talking about. Just for fun, I checked my standard push up "pace" to see how quickly I rep them out. 75 in 45 seconds with good technique. Had I done more, I would have had to slow down, and more than another five or so would have resulted in bad technique. NEVER settle for bad technique. It is common, and usually acceptable, to have less than perfect technique when going for max reps with any exercise... acceptable technique is not necessarily the same as perfect technique. There is a pretty distinctive line between acceptable and bad... don't cross it. Tip: when you are first starting with push-ups, (or any exercise) ALWAYS go slow. Not because it is a better workout, but because it allows you to focus on technique and develop proper habits from the get-go.
Chest vs. Triceps/Narrow vs. Wide Grip:
Just as with performing a bench press, keeping your hands farther apart works your chest more, where keeping your hands shoulder width primarily works your triceps. If I do two sets of push-ups, I try to do a set of each. If I am just shooting for max reps, I go somewhere in the middle. Once again, there is no right or wrong... this is based on personal preference and the muscles you are trying to work.
Range of motion:
Up
I would hesitate to say that it is incorrect to lock your elbows when at full extension, but I do recommend that you do not. I prefer to extend as far as possible without locking any joints. I have no scientific basis for this... it is based solely on how my arms/elbows feel after working out.
Down
At the lower end of the rep, MOST people should not touch their chest to the ground. This is all about angles... when doing a push-up, your spine should stay straight. As you raise and lower yourself, the angle your body creates with the ground changes. If you are indeed keeping your entire spine straight, your chest will NOT be the first part of your body to touch the ground. Depending on your specific anatomy, this could be your chin, nose, or forehead. For me, it is my nose. My goal for any given rep is to get my nose within half an inch of the ground. I don't try to touch the ground with each rep, (although I frequently do) as I would undoubtedly end up with a broken nose and a humorous emergency room story. The exceptions are individuals with larges chests... be it those who have developed large chest muscles, have large deposits of fat on their chests, or well endowed females.
Your butt:
One of the most common push-up mistakes is allowing your butt to drop or stick it up higher than the rest of your body. As I mentioned before, you should keep your spine completely straight. Because of anatomical differences, this may mean that it is "sticking up" for some people. As long as your back is straight, you are correct. There are variations of push-ups that call for these positions. Holding your butt in the air changes the angle of your arms to your body and allows you to work more of your shoulders, and a slightly different section of your chest. This is similar to doing an incline press. So... I guess that means that "butt angles" can vary, but this is more of a specialized thing and wouldn't really count as a true "bread and butter" push-up.
I have always wanted to be able to do 100 push-ups, and have been working on this for awhile now. I am pretty close, but haven't quite made it there. I formerly worked out intensely and often, but now only really do push-ups and curls for upper body. I am exactly 6 feet and weigh about 170lbs. Based solely upon my own experiences, I find it very believable for someone who strength trains regularly to do upwards of 200 reps.
I don't really want to see this turn into a contest, but if you really want to compare you should set a specific tempo for reps. A metronome would be ideal, but I know most people don't have one. You could always use an online metronome, but where is the fun in that? A good alternative would be to pick a song with an appropriate tempo from youtube and then everyone can do reps to the beat.
*cough*Let's use this one...
There is time during the intro to start the video and then get "into position." Starting in the down position: up on beats one and three and down on beats two and four. If you rep for the entire song... you've done about 130 reps.