Construction is fundamentally a business of optimizing for minimum cost within the law. Taking the foreign companies and making them operate in NYC would get you the same result as the local companies because the root problem is decades old laws that more or less serve to create regulatory capture to benefit the members of various unions and trades organizations.
What do you mean? What would give foreign companies an advantage? If a transit construction project can be done at lower cost, isn't that a good thing, for the local government and for riders? And construction is something that hires local workers anyway.
If the foreign companies can bid lower and with faster schedule, then they have an advantage to win the contracts over US firms
Foreign firms do bid, just not with lower costs or faster schedules. Skanska from Sweden is in charge of NYC's East Side Access, which was originally forecasted to be completed in 2007 at a cost of $4B, and has morphed into the world's most expensive per-mile rail project at $12B with a completion date in the 2020s.
Construction workers are generally hired locally, and I doubt that specialized construction skills would be considered qualifications worthy of priority visas in the US.