Is there a timing window between operations played by Accelerated Diagnostics where paid abilities can be used?

For example, you have a Jackson Howard, a Power Shutdown, and an Accelerated Diagnostics in hand.

• Click 1, install Jackson.
• Click 2, Power Shutdown your entire R&D into Archives.
• Then use Jackson to handpick 3 operations (Interns, something else (A), Accelerated Diagnostics) to go back to R&D.
• Click 3, play Accelerated Diagnostics and play those three operations:
• Interns to install another Jackson.
• Use Jackson #2 to stack 3 more operations (Interns, B, Accelerated Diagnostics) on R&D
• Play A
• Accelerated Diagnostics #2:
• Interns to install final Jackson
• Play B
• Use Jackson #3 to stack 3 more operations (C, D, E)
• Play Accelerated Diagnostics #3:
• C
• D
• E

No, there's no window for that. Look at your timing chart. Paid ability windows are BETWEEN clicks and at certain times during a run. Also start and end of a turn. You can't interrupt the resolution of the click you spent to play a paid ability (unless that paid ability is a prevent or avoid effect).

While the crazy combo won't work, there is nothing preventing you from installing JH on click 1, Power Shutdown on click 2 trashing everything. Using JH to put back a SEA source and 2 Scorched Earths and click 3 accelerated diagnostic for the win (assuming you have the money to pay for it all).

As we say in St. Louis, "Cheap Cheap Fun Fun"

Edit: I think I just named my next deck.

Do you play the cards as you draw them or draw cards and play them in any order? If you play them as you draw them the Sea + Scorched combo is one in three, hardly something you'd like to base your deck around.

No, there's no window for that. Look at your timing chart. Paid ability windows are BETWEEN clicks and at certain times during a run. Also start and end of a turn. You can't interrupt the resolution of the click you spent to play a paid ability (unless that paid ability is a prevent or avoid effect).

So where does using NACH fall into that? Or world NACH just not work in that case?

NACH is an Avoid ability. Prevent and Avoid effects can interrupt (and are the only things which can).