Miyabi is being really silly with this project. He doesn't even know how to build the engine for the locomotive and he's aiming for a train? He should be working on a car instead; far less horsepower needed and can travel on the existing road infrastructure. It will also fulfil the intended purpose of enabling two adventurers to travel more quickly over the 600 km distance.
Beyond the myriad of issues that come with such a project (permission to build, maintenance cost, etc), it seems to me that it will attract attention to his skills and he may be "asked" to keep working on such projects to link up other cities because he'll be able to do it in a fraction of the cost and time that it would take otherwise. Changing the travel time from two weeks to 12 hours is pretty transformative for the economy and the military after all, so there's no way the nobility and royalty won't ask him to contribute further. Then again, there is apparently a guild of infrastructure and production that manages such matters, so he'll probably be shielded of such work by the guild taking on the requests to build that type of infrastructure.
Absolutely not.
There is no existing road infrastructure. They're just plain ground, very uneven, with varying width's. He wouldn't be able to go much faster than in a horse-drawn carriage, just that he could run it for 14-16 hours a day instead of 8. He'd cut the time from 2 weeks to 1 week, which is still completely useless to them.
Not to mention, he'd be susceptible to monsters, bandits, risk endangering other travelers or outright getting stuck.
If there are any bridges along the way, they could possibly fail under the load as well (though he could make sturdier ones, I guess).
He also clearly had described how he intends to make the train move. There's literally no need for him to know how a train in our world works. No one also said that he plans on making the gargantuan things that we have currently. Who knows, it may be just the size of five or so wooden horse carriages (and may be wooden itself), so it wouldn't weight even as much as a single train wagon...Heck, depending on how it was constructed, it could even weight less than some of the larger cars. Though I do realize he probably will (that's the standard of fiction).
As for the remaining two arguments, they seem lacking.
First, permission to build? The existence of such a thing isn't as obvious. While 'building codes' are ancient (literally, apparently about 1770BC, so close to 3800 years ago), permits to construct something? And outside of a city too? While there were people that simply weren't allowed to have possession of their own, landowners generally could construct things rather freely. The codes were generic, very imprecise and likely building on your land, likely you didn't need to ask anyone. Just that if you didn't follow the building code (whatever it was in a given region), it'd get razed.
Towns and their general area are likely more strict though, so it is reasonable to expect that the construction would need to start some ways from both towns.
Second, he already attracted attention of the DUKE. At this point, him doing this isn't putting him in any more danger than he already is. If the duke wants to use him...he already will. If he doesn't, then he will be able to cover for him.
Even if not...sorry, but yeah. He's literally the last person that can actually be threatened like that. What will they do? Keep him at spears edge 24/7? Remember that there's no physical barrier that he cannot get rid of. He can just hide underground and travel literally anywhere he pleases unimpeded. It's impossible to lock him up. Even chains will do nothing, he can just get rid of them.
If this wasn't a light-hearted manga, I would say that he actually has the ability to singlehandedly destroy entire countries. Collapsing structures, sending people to their death at spike traps or crushing those around them by creating heavy items above their heads? He can do all those things. Unless you are aiming to kill him and manage to do it before he knows about the attempt...there's basically nothing that can be done for a user of magic that can do things like he can.
Of course...for plot purposes he won't be utilizing his abilities to the fullest. No protagonist ever does.