Is the nuclear engine thrust only usable from the orbit forward or could it be used in the first escape stages? I don't understand the engine's lifecycle and how it combines with other rocket stages.
Can the engine produce the peak power needed at liftoff? Or are its main benefits realized over the course of the trip instead?
Nuclear rockets are almost exclusively proposed for use as upper stage engines or for interplanetary transfers. One major reason for this is "no one wants their launch complex to glow", which is a major downside of the currently feasible nuclear rocket engines with >>1 TWR. (Orion drive, which is throwing nukes out the back and riding the shockwave, and the Nuclear Salt Water Rocket, which is like that but continuous via a standing nuclear detonation in streams of enriched Uranium salts in water)
Once you're in space, ISP matters a lot more than TWR, and is the main limiting factor on how far you can get with a given mass fraction. Nuclear rockets generally have ISPs at least double that of the current best high-thrust engines, hydrolox. (hydrogen+liquid oxygen)
Its really a shame - I've read somewhere Orion is actually even more efficient in an atmosphere! Oh well, we can still use that somewhere else where the atmosphere is not breathable or worse (Jupiter, Titan, Venus, etc.).
A nuclear thermal rocket works just fine in atmosphere. It has a similar profile to chemical rockets, in that you can tune the reaction or the bell nozzle to work well in atmosphere or in vacuum. The biggest problem with using this as a first stage is dealing with the legal and social logistics of running a nuclear reactor that isn't yet on an escape trajectory.
A solid core NTR exhaust might even be rather clean. Liquid or gas core NTR, not so much.
And if someone is planning to launch a nuclear salt water rocket inside the atmosphere of an inhabited planet, you certainly have bigger issues to worry about.
No, nuclear thermal rockets could be launched from the ground. That was the plan with NERVA, which was a pretty serious attempt to develop the technology in the early seventies. A nuclear thermal rocket engine 'looks' like a rocket engine, but there's energy from fission in the combustion chamber.
NERVA was targeting thrust to weight ratios of <5, though - which isn't high enough to lift off of the ground with. Makes a fantastic upper stage engine, but a rather useless first one.
And more specfically, it was being a proposed as a replacement for the S-IVB, the third stage of the Saturn V. Early maps (like the one shown at https://www.hq.nasa.gov/office/pao/History/SP-4204/ch11-8.ht... ) show the Nuclear Assembly Building, along with three pads, at Launch Complex 39, which launched the Saturn and Space Shuttle, and now launches the Falcon rocket.
Can the engine produce the peak power needed at liftoff? Or are its main benefits realized over the course of the trip instead?