Getting IDs from t-SNE plot?
If you are using sklearn's t-SNE, then your assumption is correct. The ordering of the inputs match the ordering of the outputs. So if you do y=TSNE(n_components=n).fit_transform(x) then y and x will be in the same order so y[7] will be the embedding of x[7]. You can trust scikit-learn that this will be the case.
Quite simple, 
If I perform t-SNE in Python for high-dimensional data then I get 2 or 3 coordinates that reflect each new point. 
But how do I map these to the original IDs? 
One way that I can think of is if the indices are kept fixed the entire time, then I can do: 

Pick a point in t-SNE
See what row it was in t-SNE (e.g. index 7)
Go to original data and pick out row/index 7.

However, I don't know how to check if this actually works. My data is super high-dimensional and it is very hard to make sense of it with a normal "sanity check".
Thanks a lot!
Best,