Lets start by assuming the balloon stays the same size as it rises in the air column and we'll ignore the temperature drop. The pressure and density of the gas inside the balloon remains the same, but at some point the air density outside the baloon will drop to match the density of helium inside the baloon. At that point the balloon would stop rising as the weight of the atmosphere it displaces is the same as weight of the helium filled balloon. It's like a little boat on a sea of air.

However, balloons don't stay the same size. As they float up the atmospheric pressure drops. The balloon will expand because the pressure inside the balloon is higher that the pressure outside. It still has a bouyant force on it because the weight of atmosphere it displaces is still larger than it's own weight, so it continues to go up. Outside pressure continues to drop. Balloon continues to grow. Eventually the balloon bursts.

Close, but now you come into contact with the atmosphere not actually being the same density (in weight/volume as well as in particles/volume) throughout, but instead gets thinner as you get away from the earth.

For simplicity, assume space is actually empty, and the atmosphere gets thinner linearly up until x kilometers above sea level it's completely empty. Then the density will also decrease with height, and the helium balloon will eventually find a spot that matches its density, and stop there.

Again there's so much more to it but as a simplified model this works 😅

Rockets mostly need to fight speed (of the earth revolving around the sun), and indeed in our atmosphere speed means friction, but in space rockets still need a lot of propellant to change their trajectory. As always there's a relevant xkcd: https://what-if.xkcd.com/58/