Is Euclidean TSP strongly NP-hard
Yes, Papadimitriou in his "The Euclidean Traveling Salesman Problem Is NP-complete" paper reduces from Exact Cover to Euclidean TSP.

All the coordinates of the points are of polynomial magnitude.

And the required precision for the $L_0$ value in page 243 is to be able to distinguish between $a$ and ${(a^2+1)}^\frac12$. And near the end of page 241, he wrote "An adequate value of $a$ is 20." So the required precision is finite. Since you only have $n(4a + 13 + 2^\frac12)$, that would only magnify the difference.

All the page numbers above are in the proceedings containing his paper and can be seen once you have downloaded the paper.