Ok, so here's where the USDA and experiential data have a giant cage match and everybody comes out bloody, beaten and hating each other.

The food safety guidelines provided by the government, and designed to make sure that food served by restaurants is as absolutely safe as completely possible without requiring irradiating rigs in every prep area of every eatery in America (and, consequently, creating a licensing boondoggle with the EPA), state that most prepared foods that are not in the extreme "basic" or "acidic" pH levels must be stored at a temperature above 140 or below 40 degrees Fahrenheit; with the range in between colloquially called "the danger zone". Bacteria is known to thrive in that zone, and cross-contamination problems (processing a head of lettucuce on a cutting board previously used to quarter a chicken, without the application of chemical sanitizers and a good wash) will exacerbate the problem. Foods that are cooked must be transitioned from the 140 to 40 degree level before storing, but the time frames vary by food type and acidity.

As a home cook, however, I can attest to leaving things like cooked chicken wings on the counter for, literally, days at room temperature without any ill effects whatsoever. Is this wise? Not really. But given proper precautions like a sealed container that is opened only to grab a portion to eat, highly acidic flavorings (traditional Buffalo wings have a high acid content due to the vinegar) and above-average cleanliness in the kitchen (I'm a home-brewer, so I keep a spray bottle of StarSan prepared at all times to "touch up" during and after cooking), you'll probably be alright.

Note the operative word there is "probably". There's little doubt that sometime, as I roll the dice with questionable food safety, that the house will win and some nasty, malevolent critter will take hold, despite my propensity for sanitizing ALL THE things, and I'll pay with a few days of violent bathroom visits for both the anterior and posterior regions.

Given your relatively short (3-ish hours) time frame, you're as OK as you're likely to be without strictly following food-handling guidelines, but there's always a chance... How much you're willing to risk is your call. I'd place it at a 100-1 bet, or so; the USDA would factor it as "you're going to die bleeding horribly from every orifice, and here's your fine." Your call.

And bonus points for not putting hot things into the fridge; the energy required to cool them is typically under what a chill box can provide, and this potentially raises the temperature of everything in your fridge into the Danger Zone, increasing your chances for bacterial hell.

Normally, you're ok for a good amount of time after cooking. However, ground beef is a completely different beast altogether. The process of grinding the mean ensures than any surface bacterial contamination of meat is homogenized throughout the ground meat. If you didn't cook them to a sufficiently high temperature, you've let a possibly large amount of bacteria survive and multiply.

Grass-fed beef doesn't mean it's some magical creation that cannot go bad over time.

Why would you have leftovers? :)

If you are really worried, I would crumble them up and fry them with some scrambled eggs. Frying them again should kill anything bad.

The Navy's food safety rules said (when I learned them, some 19 years ago) that food couldn't stay in the danger zone (40-140F) for more than 2 hours total before it had to be discarded. Not knowing if you went for rare meat or cooked it more than that (the more cooked, the higher it was than 140 to start with), I'd say that you're probably close enough with the 2-1/2 hours on the counter that you could still eat them.

I have heard that putting hot or very warm meat in the fridge can cause bacteria (or some other nastiness) to grow, so I'm with you on leaving things out until they cool a little bit.

On occassion I've accidentally left things out overnight, usually meat, and I've never felt any ill effects.