Spica

Yes, each successive chamber needs to contain significantly hotter gases than the preceding one, but at that point why not make the first chamber larger and hotter. The trouble here is bridging the gap between chemical (thousands of kelvin) and nuclear (hundreds of millions of kelvin) explosions. You could use an electrical discharge instead, but then you ought to check whether an electric light gas gun is more or less efficient on an energy basis than a laser rocket like this: Such a vehicle would also be FAR gentler to any payload it carries than a gun-launched vehicle would.

Unfortunately this trick doesn't actually let you gain significant speed beyond what you'd get from a normal gun. Once the projectile reaches speeds close to the practical limit of a single chambered gun, any additional chambers provide little if any further velocity increase. Think of it like this: before the gas from the additional chambers can push on the projectile it must first catch up with the projectile. Once it has accelerated to match the speed of the base of the projectile little energy remains to continue to accelerate the projectile along. A more technical description is found here: https://www.researchgate.net/publication/268456817_A_comparison_of_distributed_injection_hypervelocity_accelerators There are other tricks that can work though, which are described in more detail in the paper I linked here. One of which involved injecting gas perpendicular to the barrel and expanding it along a tapered projectile.

Well this is basically the textbook case of "the devil is in the details". In order to build a working rocket engine you need to figure out exactly how to get the propellants into the engine, mix them, ignite them, and send them down the nozzle. Here are just a few of the problems you need to solve to make a working rocket engine: The pressure in the tanks is lower than the pressure in the engine, how do you make sure the propellants only flow "uphill" from the tank to the combustion chamber? The gases in the combustion chamber are burning at temperatures of over 3000K, how do you stop the engine (this includes any fuel injectors or ignition system too) from melting or otherwise falling apart? You need to control it, how? With what? Remember, the precise shape of every part much be chosen to make it as light at possible, what material is strictly necessary? Where isn't it necessary? Here's an unofficial cutaway diagram of the raptor engine's systems, each one of these solves some problem mentioned above, along with countless others that aren't mentioned.

I remember there was some lively discussion about the new flap configuration when it was first announced, but I think it makes the ship look so much more put together.

There's one big snag with reusable nuclear rockets that I don't seem to see mentioned often enough, especially in discussions of reusable nuclear first stages: A nuclear reactor that has never fissioned is not going to be radioactive to any worrying extent, but that all changes after it's first turned on. The more energy it has generated throughout its history, the more radioactive it is. This will make each successive re-use more difficult and dangerous than the last, since a reactor light enough to launch a rocket with will not be heavily shielded. If the reactor is not submerged in the ocean during the recovery process don't expect crews to access the ship without exceeding their legal yearly radiation limits in a matter of seconds. Here's a diagram of a NTR powered reusable nuclear shuttle concept from the '70s, showing the radiation levels after engine shutdown. High flight rates are not going to be possible without highly if not fully automated reuse procedures.

Triton gets about as close to zero as you'll find in the solar system, with an orbital eccentricity around Neptune of 0.000016

This right here is 96% steam by mass. The thing you see in daily life and call steam is nothing compared to "thermodynamically interesting" steam. The transparent part of the plume is around 1200 K, and the pale blue Mach disk is around 3,200 K.

You absolutely can, all you need to do is attach a rotor that's twisted the other way to the same vehicle. In that case, the rotor will act like a turbine that slows the air in the car-fixed frame, and uses that power to drive the wheels.

If your goal is to try out all kinds of propellant combinations, you should also give RPA a look: https://www.rocket-propulsion.com/RPA/download.htm. The 1.2.9 lite edition is free to download and use. I've also found it a good bit easier to use than CEA for rocket engine analysis.

A not completely unreasonable guess would be to assume a burnup threshold beyond which the reactor can not be started, due to a buildup of fission products. I really have no idea what that usable fraction of fuel energy is, and it will surely depend on the reactor design. Let's use 5% as a value for now. 1 kg of fissile material contains around 80 TJ of nuclear energy within, and with a 5% usable burnup that gives us 4 TJ/kg. Hydrogen takes around 44 MJ/kg to be heated to 3,000K, a warm but not unreasonable NTR core temperature. This means that 1 kg of fissile material provides the energy to expel 90,000 kg of hydrogen propellant, that's a lot. This leads me to think that the limit is not one of time, but one of "propellant expended", and it will be fairly long. But beware the safety hazard that large amounts of long lived fission products create, I wouldn't want to be caught near one unshielded, and the shielding will be thick and heavy (and usually just a puck above the engine, since only the ship that carries it needs to be shielded).

Conservation of angular momentum prevents you from strictly stopping your own rotation purely with internal processes. You can redistribute any angular momentum within your own body by whirling your arms or using a gyroscope, but you can't ever get rid of it by yourself. Of course if you can throw things away from you or react against some external object or field you have more options, like RCS or a magnetorquer.

Just to give you an idea of the energy scales required for this task: Accelerating 2,000 tons up to 8,500 km/s with a rocket requires at minimum 1.11*10^20 Joules, or around 25 Gigatons TNT equivalent. This is with a propellant mass ratio of 4.922, and negligible structural (non payload) mass. The power required to perform this task is also on the order of 140 kilotons per second. This task can be performed with Uranium as the energy source, but it will require about 1300 of your 1 ton modules, and that's just for the energy requirement, you'll need even more to account for the propellant you need to do this. If instead for example DT fusion powers this, the "radius of instant death" for unshielded people is on the order of hundreds of kilometers.

Starship is not designed to ascend at supersonic speeds under it's own power. I assume it is both unstable and insufficiently controllable in that flight regime without being attached to the superheavy booster.

I see three issues with silane as a choice of propellant for ISRU on the moon: 1) I assume any engine intended for this role ought to be reusable, and this seems to conflict with the fact that the exhaust of any silane fueled rocket is largely quartz, some of which will be left behind in the engine after shutdown. Solid silicon deposits may also build up in parts of the engine plumbing wherever high temperatures are experienced. The large amount of solids and liquids in the nozzle will also reduce Isp. 2) Hypersonic quartz smoke in the exhaust will be damaging to any nearby structures on the ground and in orbit. 3) You still need large quantities of hydrogen in order to make large quantities of silane. I'm not sure how important each of these problems would actually be in reality, but they certainly need to be either solved or shown to be insignificant in order to make silane a useful propellant.

Try applying Newton's third law to this problem, you'll find that this is impossible for real matter. Newton's third law and momentum conservation come hand-in-hand, and there's no way you'll be able to violate either, in any situation.