You can't make a warp ship out TITAN II missle with 21 century Technology of Star Trek.
Why not? People seem to be under the illusion that 21st century tech in Star Trek is somehow similar to 21st century tech in the real world. It's not. We have had interstellar probes for a decade now, artificial gravity for two. Why on Earth should it be even moderately difficult to build a starship out of garage junk when that junk in all likelihood consists of leftover antimatter batteries and flying belts?
[warp core] was supposed to be new tech for the Galaxy and onwards
Huh?
Nothing in TNG was supposed to be new tech, save perhaps for holodecks. And even that "perhaps" was later clarified when it was revealed that the civilian world had operated passable holoentertainment devices for several decades before "Encounter at Farpoint" - thereby establishing continuity with TAS "Practical Joker".
M/AM was also supposed to be relatively new tech for the TOS era
Again, not.
Nothing in TOS was said to be new technology, save for the M-5 multitronic computer and its predecessor technology duotronics. Phasers, transporters, warp drives, shields... They were all old news. If some technology was considered "old" and not up to the hero ship standards, it was
centuries old, as in "Where No Man Has Gone Before" or "Balance of Terror".
It's not possible to nail down a timeline of technological breakthroughs from TOS and TNG and then claim that ST:FC somehow lies ahead of that logical curve. Every bit of relevant Trek technology was always developed well in advance of the respective shows and movies. ENT was basically the sole exception to this, its premise being that a new type of warp drive had just emerged and made mankind's first real deep space exploration mission possible.
Kirk and Picard were handed their superior technology on a platter. Why should we think Cochrane was not?
TOS explicitly said Earth did not have the nuclear holocaust
Where?
Of course, if the Phoenix had an intermix chamber, then it had to be a M/AM reactor...
Of course not. Saturn V had an intermix chamber, too - in fact eleven of them, 5+5+1 in three stages. "Intermix" is not inherently related to antimatter in any way.
Why not? Antimatter power could well be commonplace in the 2050s of Trek Earth. There's absolutely no reason for it
not to be - it's purely up to the writers of the pseudohistory to decide whether Earth in a fictional alternate history operates antimatter batteries, Mr Fusion domestic power stations or perhaps cosmic energy taps. The writers of Trek have chosen antimatter, and have not deviated from that path.
There are major corporations who have blown millions of dollars and multiple test flights trying to figure out how to recover a rocket stage at sea in the manner you describe. All of these organizations have far greater resources than Cochrane's team, more money, more infrastructure, and every one of them depends on the cooperation of NASA--using their tracking network and recovery ships--for the recovery effort to be even SLIGHTLY feasible.
Why quote organizations that are at least a full century behind Trek in technological prowess?
We can see with our own eyes the fantastic performance of the Newtonian (or perhaps non-Newtonian?) engine that pushes the
Phoenix to outer space - a rocket a thousand times more powerful than anything we have ever even dreamed of. It would be trivially easy to use that fantastic drive to simply
hover down to Earth's surface if that were necessary. With power like that, one can forget all about ballistics; if it amuses one, the ascent path to orbit can be the shape of one's signature, the escape velocity perhaps two miles per hour.
Of course, nothing in the movie indicates that Cochrane would even want to return the warp engine down to Earth. If it's the most valuable piece of hardware in the history of mankind, surely he'd want to keep it out of the greedy hands of just about anybody? Instead of pondering the engineering details of landing, Cochrane would spend his time well installing a powerful scuttling charge and then telling his customers that the bidding for the currently orbiting piece of technology starts at one trillion, with his finger on the big red button.
The winning bid would then result in the buyer being given the permission to pick up the warp engine with his corporate or personal space ferry, supposing he or she didn't prefer keeping it in orbit and preparing it for another test flight.
As for the Bussard collectors, I again remind everybody that we don't have the faintest idea of what they do in the Trek reality. We only know their name and their ability to belch out hydrogen or scoop in explosive gases for combat use.
In the real world, we know that a technology of that name was once considered for interstellar propulsion until it was found out it doesn't work, except as a huge drag chute (there's always more drag from it than can be countermanded by rocket power derived from it). We also know from the real world that names get recycled once the original meaning falls to disuse.
And we know that those glowing red things are present in virtually all Federation warp drives, suggesting they are a vital piece of technology rather than an optional extra.
Timo Saloniemi