There's definitely a lot of interesting ways to go with railroads as a form of "modern" travel. I do think it makes some sense to have you build it using a charged unit, as that would be a decent enough cost to prevent them from being spammed everywhere. And I agree that having movement essentially be a form of "airlift", where as long as you have an unbroken path of railroads, you can move between those cities. Maybe you could also have a cap of 20 moves so that you can't move from Paris to Johannesburg in one turn by rail, but that would also limit where you needed to spam them (as it would only matter essentially whether your cities were connected or not).
I'm not sure if using it as a full district would make sense, as district spots are probably a bit scarce. But maybe it would just be another city centre building (railway station). I do like having some other benefit for having your cities connected by railroad - maybe even something as simple as if two cities are connected by railroads, the trader will simply instantly transport between the cities, and you also get +25%/+50% to all yields of the trade route?
I'd vote for a railway station as a city center building and as a tile improvement. Building one automatically makes tracks 'appear' connecting it with any other stations within maybe 6 tiles?
Definitely should be trade bonuses, and maybe production bonuses ala civ 5.