If I am cooking steaks at a grill and one of them accidentally falls down on the floor/ground, is it safe for me to wipe it and cook it for a certain amount of time (till it is pretty well done) and then eat it? Or should I throw it away?
This depends completely on the context.
Are you at a grill in let's say.. Outback Steakhouse? If so, please throw it away.
Are you at a social event or home cooking for yourself/others? clean it off with water and you're good, maybe even feed it to someone you don't like afterwards (unless it's the biggest and best steak, then you gotta eat it.)
It depends on what it is likely to pick up.
If it's an interior floor surface that is usually kept clean, you're unlikely to suffer any ill effects.
If it's outside on the ground next to the gas grill and you regularly fertilize/pesticide/herbicide the lawn, or sealed/stained your deck/concrete recently, or have a number of animals that use the space as a restroom and play area, I'd suggest throwing it away and being more careful with your food.
I don't believe one could safely answer, "Yes, it's generally ok regardless of other factors."
That said, washing it off after contamination, and ensuring it reaches safe cooking temperatures after washing it will probably prevent most significant opportunities for poisoning or illness.
Whether it's worth the risk depends entirely on how risky it is, and how risk averse you are.
5 seconds under the tap, then back on the grill. If you have any organisms on the floor / ground etc. capable of surviving proper cooking of meat within 30 seconds of being exposed to said meat, your guests are doomed. ("The Salmon Mousse!") They won't be on warm meat long enough - E.Coli (the bad one) needs 20 minutes to divide, and that's after a zero-growth lag period.
To take this to a logical extreme, rubbing your raw, wild rabbit medallions (that you shot out of season and butchered 30 minutes ago) on a nearly-dead ebola victim's face and then cooking it medium-well will still produce edible food. Wash your hands before sitting down at the table, please.
However, as others have mentioned, inorganic chemicals will not be affected so you could have taste or toxicity issues. I would hope that your kitchen floor has not been dusted with arsenic powder, nor that you are barbecuing outside mere hours after the people in moon suits sprayed your yard (and hopefully removed the ebola victim).
If a steak hits my deck, I rinse it with the garden hose and return it to the grill. That's if no one is looking. If people are watching it goes into the kitchen, gets rinsed there, trimmed so it looks different, and then the "replacement" goes back on the grill.
If you work in the open grill at Outback, make a big show of the meat's disposal and floor cleaning.
A caveat to the washing off suggested by others, I would add:
Do not wash so thoroughly that the flavor is lost - you may as well throw it away then. Just pour water for 3-4 seconds. After that, if you feel it is still dirty, then take a serrated knife and thinly scrape off the part that touched the ground.
It is safe if you take some measures to clean the dust and other particles that might have sticked to the meat.
Rinse the meat with lots of water, no soap. If you have some sauce to spare, after rinsing the meat, set apart some sauce (throw away that sauce afterwards) and use it to season the meat throughly. This will remove almost all the particles and substances that the meat picked up, and re-add the spices the meat lost.
After that, grill the meat again. According to the Fodsafety.gov, temperatures above 165F are enough to kill the germs. And this link shows that most grills have a surface temperature over 300F anyway.
So your steak will be safe to eat, and if you season it well, nobody will tell the difference.
Even if you don't cook the steak afterwards (that is, assuming it was properly cooked when it hit the floor), keep in mind that the human immune system is more capable than we give it credit for. Moreover, pathogens that do make us ill, such as E. Coli and Streptococcus, prefer moist areas with an easy supply of nutrients; our floors and counter-tops aren't typically their first choice for an abode.
As already mentioned, I would probably rinse it off to get rid of the worst, as well as remove anything that may affect the final taste, but as my Swedish father always says, "Lite skit rensar magen."
However it's proven false, i still would like to mention the following:
In our country we apply the 5 Second Rule, which is very populair.
It basicly comes down to this: Whatever kind of food you drop on the floor, if you pick it up in 5 seconds, it's ready to go!
Never had any complaints ;-)