Great write up, thanks. I'll give them a try
A few things if I may:
1. When you design a stackup, make sure you use core and prepreg thicknesses that actually exist.
2. If you can, route the most critical hi-speed traces on the core (so that one side would be the signal layer, while the opposite one will be the reference plane to which these signals are referenced). The reason being is that core thickness is more stable and predictable, unlike prepregs, which final thickness tend to wary a little during lamination depending on a "fill factor" of a layer it's been laminated to.
3. Include a text file in you archive with gerbers which would list all requirements, explain what each file in the archive is for, desired stackup, requirements for controlled impedance, and anything else you think might be useful for them to have in order to manufacture the board the way you want. If you want, I can post the file I've sent so you will have a sample.
4. Once they complete pre-production step, they will send you production files for your review and approval, a document with field solver sim results that will show what impedance your traces will actually have taking into account materials they are going to use, as well as any questions CAM engineer might have. This is your final chance to verify that everything is correct, but keep in mind that these gerbers are the actual files that they will make photo masks from, and so traces on them are going to be a bit wider than what they should be (to account for some over-etching) - and consequently spacing will be a bit smaller, but this is how the process works and should not cause any problems. Again, if you're curious what this looks like, let me know and I will post the files I received so that you will get an idea of what you're up against.
5. To verify that the layers order is correct, you might want to include this kind of thing in your design:
Here I have a layer number in each copper (each layer only has a single digit with layer number, and they are offset so that you see them next to each other) with soldermasks removed from the area, so that when you have a board in your hands, you can easily see layer order by looking at this area in bright light - the deeper the layer is, the more faint the number appear. This is a trick that I've seen in some youtube video, and it allows you to easily and non-destructively confirm that layer order is correct, so I now always include it in all my designs as a sanity check. Not that I don't trust them (so far I don't have any reason to doubt their professionalism), but we're all human beings, and human beings do make mistakes every once in a while, so this small element will prevent a major headache to debug the board only to find out that the order is messed up.
For 6 layer boards, their production time was 6 business days since I approved production files in both my orders. Preproduction step took 1 or 2 days in my experience from the time I paid the invoice.