Space travel using 2 stage high-altitude balloon + vacuum

No. Say air at room temperature & pressure weighs 1 lb per X cubic feet. We'll make a balloon with volume X. Further, assume the 'skin' of the balloon has negligible weight.
Lift is the difference between the weight of the balloon (which is just the weight of the gas within) and the same volume of air.

I'll just "make up" densities of other gasses so you get the idea.
AIR density 1 LIFT 1 - 1 = 0
HELIUM density 0.1 LIFT 1 - 0.1 = .9
HYDROGEN density 0.05 LIFT 1 - .05 = .95
GOOD VACUUM density 0.0001 LIFT 1 - .0001 = .9999
PERFECT VACUUM density 0 LIFT 1 - 0 = 1

Helium is almost as good as hydrogen (and a lot safer)
The degree of vacuum hardly matters and you need a thick & heavy structure to prevent your 'bubble' from collapsing like the hull of a deep-diving submarine.

Again, no.
Slingatrons (and catapults in general) haven't been used for warfare since gunpowder was invented.
Guns and rockets give much better performance and weigh much less.

Yes but at high altitude the equation for vacuum balloons would be a bit different right?
Outside pressure would be less, so a lighter structure or better vacuum could be achieved

Lighter structure, yes. Except that you had to have it to survive the initial lift from the ground.

And if the external air is half as dense as it is at ground level, the lift of even a perfect vacuum is halved.
Give it up. It doesn't work!

Parenthetically, I read an obscure SF novel once where spaceships had forcefields. These were expanded, creating a bubble of vacuum around the ship. Continual enlargement enabled them to rise to the top of the atmosphere -- whereupon they flicked though hyperspace. Clever notion, but we don't have forcefields.

Wouldn't help, look back at it like this you're launching a rocket at some height [math] h [/math] above the earth, so neglecting some constants we see that [eqn] \frac { 1 } { r^2 } - \frac { 2h } { r^3 } [/eqn] now since [math] h \approx 10 [/math ] and [math] r \approx 10^3 [/math] then we're only reducing the force we need to work against by something on the order [math] 10^{-2} [/math] so a lot of work for practically no gain.

Gun muzzle velocities are limited to the speed of sound through the gasses pushing the projectile.

If you use an electromagnetic gun you're not limited in this way and can reach much higher velocities, but the most powerful rail guns we've ever constructed on Earth only achieve about half of orbital velocity.

>However, balloons have been used to lift rockets beyond the denser layers of the atmosphere before igniting them. This is practical and there are plans to use this method to orbit microsats.

I agree, but they're only really feasible for very small payloads. You won't be building space stations or anything else requiring more than one ton to orbit using rockoons.

This idea and ideas like it are silly. They get the problems associated with space travel backwards. Every idea proposed can't be scaled up beyond a few hundreds kilograms into low Earth orbit. They follow the idea that rocket fuel is expensive, and since these ideas save on fuel, therefore they must be cheaper and better.

In reality, no rocket that has ever launched has been more than 1% fuel by cost. The reason people say fuel costs are too high is because it's more complicated to say that going to space with a big payload requires a lot of fuel, and you need a lot of rocket to contain and push that fuel, and rockets are very expensive. Rockets are expensive because they're very complex machines operating close to the limits of materials and structural engineering. Going to space is expensive because we throw rockets away after a single use (with a couple exceptions). Therefore the cost of a single flight must cover the entire construction cost and associated costs of getting that rocket built and ready to launch.

Now you may think that the solution is to get away from rocket technology, but I disagree. Rockets are more scale-able, more capable, faster to operate, and already exist, as opposed to literally every other proposed method to achieve orbit. It makes way more sense to spend the engineering effort required to build fully reusable rocket powered launch vehicles than it does to abandon rocket technology and go for something completely different, which would never be able to achieve the up-mass capability of a rocket anyway. The same goes for any method that starts off using balloons or a plane or anything other than a rocket, then transitions to using a rocket later anyway. The initial launch phase to get above the atmosphere requires the least delta V but the highest thrust to weight ratio, so a rocket makes the most sense there too.

Not OP. And not trying to encourage OP.
The point of rockoons is/was to minimize atmospheric drag.
Reaching orbital altitude is pretty easy. A V2 could do it (and maybe Richard Branson can too.) Orbital velocity is the hard part.

>The point of rockoons is/was to minimize atmospheric drag.

As well as the thrust inefficiency at sea level due to ambient pressure. The mass limitations involved with a balloon structure limit the rocket's mass severely though, so it's pretty counter productive.

Imagine the balloon required to float a ~1000 ton fully fueled rocket in the upper atmosphere.

The answer is No.
You could bring additional un-inflated balloons with you as well as extra helium in a canister, and when you get to a certain altitude you inflate those so that you can get even higher before the balloons bust.
> But the balloons will bust
> And even if you have tubes running to them to siphon off gas, you're going to end up with neutral buoyancy at some point.

There is also an inherent lack of Velocity in this, you could maybe achieve 30 or even 35 miles altitude but you're not going to get higher and this only provides elevation.
> might make for a good way to launch pint-sized satellites from a high altitude though.

Yes I said 30-35 miles, which is a higher sight than the standard 20.

I don't know but now I hope we will some day have a space balloon. I want to be the first man to go all the way around the universe in a space balloon! When I get back to the place where I began I will have a cocktail, but only one, because drinking is for losers.