I think coding questions are really important. You see their logic flow. Now stupid coding questions (in a list, find all the number pairs that add up to another number in the list) are terrible. They're complex and even good programmers need time to think about them. Fibonacci is one that people expect, so they look up all the variations and you get people who are good test takers (would ace a GRE/MCAT) but not good designers.
You want a simple question that isn't common, but that shows how they break down a problem under stress. Example: you have an input with paragraphs at 80 characters. Write a function to return the same paragraphs wrapped to 40 characters. You cannot break a word and must maintain paragraphs.
Great design questions: a word problem (You have an autoshop with, staff and customers. Customers can own multiple cars. A staff member gets assigned to a car with a work order...) .. draw an ER diagram. This is actually a pretty low stress question. It should be straight forward. If someone draws a terrible ER diagram with lists in tables and no normalization, or unnecessary relationships (or you have to keep asking them to label 1-to-n/n-to-1 relationships and they struggle), you know they're not going to be good at designing database schemas.
Another great general knowledge question: "A user types in a web address into a web browser and hits enter. Describe what happens. Go into as much detail as you can." This gives people a change to elaborate as much as they can. People can talk about DNS, HTTP, load balances, HTTP request/response, cookies, load balancers, web apps vs static content...
Questions need to be geared to the job. You don't ask someone to draw an ER diagram if they're being hired to rack servers and setup VMWare. Likewise you don't ask a web developer to write a function to do matrix multiplication.