Please take this in the spirit it is intended, which is to help you not feel stressed.

I think the most helpful thing you can do is understand the priorities of the non-developers and speak to those points. If the non-developer believes you understand their priorities they will have confidence in you, and they will take your input/responses at face value.

If you think that your ability to communicate with non-developers is limited find an example that you can learn from to improve your effectiveness. Sales engineers are technically proficient, but they have the ability to present and communicate to non-technical parties in an effective manner. Part of the key to that is focusing on the objective rather than getting someone to understand something as well as you do.

It's not hard for all developers. I've been working with non-developers for decades, both as a developer and a development manager, and I haven't found it difficult (and as far as I know, they haven't found it difficult to work with me either).

"Non-technical people often express doubt about answers provided by programmers."

Programmers (including myself here) often give the wrong answers. A healthy degree of skepticism is a good thing to have.

"Non-technical people try to 'help' by Googling stuff and sending links..."

I've never encountered this.

"What does HN think about good strategies to avoid feeling stressed out by trying to work collaboratively with non-technical people?"

You have to remember that developers are just a small part of what makes a company work (even a tech company), and that the other people are just as important as you are. For example, without salespeople, there may not be any revenue from which to pay the developers. Without QA people, the developers' code might be too buggy to release. Without product managers, there may not be a sense of which direction a product should go. We all have to work together.