Space

Technically yes. The only thing keeping it near the object it's orbiting is that objects gravity well. The only thing keeping the orbiting object from falling into the well is its speed.

Depends on your reference frame

Adding to @MaggiWuerze@feddit.de's answer, an interesting tidbit: solar wind, particularly during more solar activity, blows (some of) Earth's atmosphere all the way past the orbit of the Moon... so at some points in time, the Moon not only falls with style, but also falls through Earth's atmosphere!

Let me first state that I'm in no way an expert nor do I know much about the specifics of what you asked. Going in with this understanding, let's do some overgeneralizing and sciencey handwaving:

Those are fantastic questions that get into really interesting concepts which I'm now going to title Types of Floating. Lemme clarify something: when I said "this level of orbit," I was referring specifically to the ISS.

The ISS and many other satellites are in what is called "Low Earth Orbit," with "low" being a relative term. If you stretched a rope around the Earth’s equator you’d have a rope that’s a little less than half of the average distance to the ISS. The orbit level of things is defined by the ranges of distance they travel around the earth, but by definition, they must go around the earth (if their orbit is defined to exist around the earth. You can orbit other celestial bodies too). So, imagine the lower earth orbits as something we lobbed hard enough into space that it'll gradually reach the peak of its height before coming back down over the course of days to years. If you want that thing to stay in space longer, then you need to give it a boost every so often by ejecting some sort of matter. For example, if you were yeeted into Low Earth Orbit and while up there said "fuck my shoes" and kicked them off your feet, you'd change how long you were gonna stay in space! If you throw your shoes forward, you'll slow yourself down and return sooner and if you threw them backwards, you'd stay longer! You wouldn't cause that much of a time difference, but there it would be nonetheless.

So, if you can leave the earth at such a speed that you'll return, could you enter space at a speed fast enough to never return? Absolutely! At a certain speed, future generations could use you as a landmark (spacemark?) on a trip out to the moon like a dead body on Space Everest. If you had been traveling any faster than that, then you could potentially fly an infinite distance away from the earth or possibly crash into the moon and have a crater named after your splattered body.

Now for the final part of your question: equating orbit to a balloon. If you're referring to an air-filled balloon that slowly falls to the ground, but gently tapping it from below makes it float around a little longer, then no. If you're talking about a helium-filled balloon that floats away until the helium escapes and it falls back to earth, then also no, but a different no. Both of these examples deal with density, but let's start with the air-filled balloon.

The reason that a gentle push on the air-filled balloon is different is because orbit is determined by an object's speed at which it circles the earth. If you want the ISS to go "higher," you give it a push forward and it goes faster and thus expands its orbit. If you want it to go "lower," then you push it backwards and by slowing down, its orbit gets closer to the earth. With the air-filled balloon, you can only keep it in the air longer by pushing up; if you blow air behind the balloon to make it move forward faster, it'll land further away but in the same amount of time.

The reason air-filled balloons fall slowly though is because they weigh only slightly more than nothing and air weighs exactly the same amount as air. This means that the mass of the balloon exists over a larger area and has a reduced density compared to a deflated balloon. In fact, I'm sure someone with more patience than I have could calculate how much air you'd have to put into a human so that they'd gently fall like a dead, morbid balloon.

As for the helium-filled balloon, helium is so much less dense than air that there exists an amount of it which can offset the rubber membrane of a balloon. (Total side-note: the lower density of helium is what gives you a high-pitch voice when you talk on helium. I take a medication called ozempic which slows down my digestion—this means that when I eat something too greasy or have too much meat, my stomach acid will break down the meat to a point that I generate a considerable amount of hydrogen sulfide. This makes my burps smell of sulfur when that happens, but as the hydrogen evacuates the air from my larynx, my burp becomes higher and higher pitched like I'm on helium.) This significantly-reduced density for the balloon means that the balloon can float on the air like you can float in a pool. As a cool experiment, you can try inhaling and exhaling while you're in a pool to note how much more you float when your lungs are filled with air. Just be careful not to breathe in water.

Anyway, you can see how in neither scenario does accelerating the balloon forward cause it to go higher.

Now, you might be wondering: when you're so far away from the earth that it's rotating at a different speed from you, how could you tell which direction is forward if you were floating in space? You could use the parallax of celestial bodies to figure out your vector, but I have no clue whatsoever how to do that. And that, the vast relative distances and speeds of objects in space is what scares the shit out of me whenever I imagine floating in space. That's why I advocate for taking care of our pale blue dot floating through space: the nightmarish possibilities of leaving earth at any amount of wrong speed or timing could lead to becoming an unknown speck floating in an unknown region of neverending space. I'm not looking to Mars for salvation with that image in my brain.