Does LINQ to Objects keep its order

Yes, Linq Select is guaranteed to return all its results in the order of the enumeration it is passed. Like most Linq functions, it is fully specified what it does. Barring handling of errors, this might as well be the code for Select:

IEnumerable<Y> Select<X, Y>(this IEnumerable<X> input, Func<X, Y> transform)
{
    foreach (var x in input)
        yield return transform(x);
}

But as Samantha Branham pointed out, the underlying collection might not have an intrinsic order. I've seen hashtables that rearrange themselves on read.

It depends on the underlying collection type more than anything. You could get inconsistent ordering from a HashSet, but a List is safe. Even if the ordering you want is provided implicitly, it's better to define an explicit ordering if you need it though. It looks like you're doing that judging by the method names.

In current .Net implementation it use such code. But there are no guarantee that this implementation will be in future.

private static IEnumerable<TResult> SelectIterator<TSource, TResult>(IEnumerable<TSource> source, Func<TSource, int, TResult> selector)
{
  int index = -1;
  foreach (TSource source1 in source)
  {
    checked { ++index; }
    yield return selector(source1, index);
  }
}