For Tali's "backstory", I have the idea of having her be Taylor's penpal or something similar. After obtaining her engineering degree or equivalent via online classes (thus making it clear her engineering abilty is not a power) due to her unique appearance, Tali started searching for a job when she heard the news of a flying cargo ship in Brockton Bay, where her penpal lives. After a few exchanged letter/emails that included her sending a C.V. and GravTech accepting it (or her being approached instead), Tali made the journey to Brockton Bay. Alas, some racist asshole 'lost' most of her things in a 'parahuman incident'. So, lacking all but her suit, Tali went for the house of her friend.
Taylor is so going to make the Millenium Falcon for when she goes to Mass Effect with Tali, isn't she? It will even have space left for her FTL drive and many other devices she will invent. Maybe Vicky and Amy gave her the idea and pushed her to make it real before the whole thing with Tali, partially out of wanting to see such a ship real (and pilot it), partially out of a desire to blow people's minds. It was simply the need for a spaceship that got her to finish the plans and present it so work could start. That and most if not all Star Trek ships that are not shuttles or fighters are just too big for two or four persons.
If they use the smaller smuggling spots (because there must be more than just those seen in A New Hope) to house some of the ship's systems and use Taylor's likely smaller systems, they then can use the freed space to make more smuggling compartments like those used in A New Hope where they would instead place workstations and storage compartments that could be lowered into the floor when unused.
Add the cargo bays (with various access methods) to store materials, equipment, components, etc. and the superior sensors and computers Taylor would invent and you are good. This way, they would bring with them what they need for their engineering space and science lab! Also, the Falcon is supposed to require a crew of two to be flown adequately, so in a way fitting for the journey of the duo. And if Amy and Vicky can join, well there is the two rear seats in the cockpit and the turrets would be available to use if needed while the other two pilot.
Thinking about it, the YT-1300 series is the kind of ship Tali (and most likely Quarians in general) would like. Reliable, easily modified and personalized to one's liking, allowing it to basically be an engineer's pet project. The amount of modifications most likely making the ship never be silent (and add to the required maintenance). And the Falcon looks like 'junk' so if Taylor gives it the same appearance, then they would not be really bothered, as was the point of a smuggling vessel by appearing unimportant.
And if Taylor equips it with Star Wars Ion weaponry along with other things like stun blaster cannons, well, it's just so they can defend themselves. It would have nothing to do with the ability to easily disable hostile vessels, forces and Geth platforms. Or the ability to disable slaver ships unnoticed if combined with the cloaking technology and leave their ground forces knocked out for when authorities arrive.
(Even if Taylor would likely give the Geth the benefit of the doubt since they did not pursuit the Quarians or resumed hostilities since the Morning War. Which is illogical to do for a enemy that can build vessels without needing to dedicate space for biological needs, can mass produce combat mechs and had centuries to do it. It would mean that the Geth do not desire the destruction of the Quarians.)
Another idea would be to invite the Star Wars Original Trilogy cast and production team (maybe invite the prequel team and cast as well while we're at it) to have a ride (in multiple groups) in the recreated Millenium Falcon. (There may be a potential future deal to lease or sell one to LucasFilms and co.) And maybe also get their help to make the recreation as faithful as possible, except the weapons for obvious reasons.
I can just see Mark Hamill asking to be the first to be allowed to pilot a X-Wing when one is made, after being taught how to of course. Harrison Ford may ask to use the Falcon for some of his flight tours with children instead of an helicopter. The Falcon (and other Star Wars designs) does come with the sound translation of sensor data for a more organic analysis when you don't have the time to check the sensor data.
PHO and other websites would go insane for a moment, before admitting that most of them have had such thoughts at least once ever since the cargo ship was moved... and how jealous they all are.
"You lucky bastards get to live the dream of piloting the Millenium Falcon or be aboard as it flew around... I am so jealous that I had to go check in the mirror if my skin turned green from the sheer envy I feel."
"Harrison Ford piloting the ship in real life must be quite the surreal experience."
[Mark Hamill] "I called dibs on being the first allowed to pilot the X-wing when they get around to make it. After getting the necessary lessons. Insert Joker Laugh here."
"Please, I am already crying of envy. Just... stop."
If the Falcon was discreetly brought to say, Skywalker Ranch and was first seen by the public lifting off there, things would go even more insane because there would be people arguing that George had it all along.
"He was being visited by Han Solo! The films are real!"
Of course, when Taylor finishes inventing a real Tricorder, I can see people begging her and the owners of the name to let the device hold said name. If she makes a Federation Starship, I can see someone asking her when she met Scotty after she started speaking engineering jargon. The same for Spock after she went on a science tangent.
If you add a 'simulation' setting like Star Trek along with safeties for collisions and crashes, I can just imagine Lucas wanting to see (and maybe experience) what it would be like to have a dogfight of X-Wing vs TIE fighter, maybe with the Millenium Falcon added as well.
"This is like playing Rogue Squadron for real."
Jack O'Neil from Stargate would be begging to be allowed to go the Earth Bet for the chance to pilot one of such crafts. Teal'c would likely simply ask if he can as he would like to try the ships out and compare them to the Goa'uld Deathglider.
Another unrelated idea would be that Taylor may ask Parahumans to participate in the research as Parahumans. For example, what are the effects of a flying Parahuman interacting with a gravity device? If a Parahuman capable of flight is provided with a gravity device that handle part of the flight, is the power used in that part redirected to the other parts? As in, if you give Vicky tech so that she float, but the device does not give propulsion, would her power compensate by redirecting the unused power to the propulsion component of her flight?
It also skips NEPEA because it is asking the Parahuman for their services because they have powers, and thus is not an unfair advantage. Anyone arguing otherwise would be shut down because it is the same thing as a job requiring two arms not accepting applicants with only one.