When sending a survey to customers, how do you ensure you aren't sending too many questions, but also covering all the things your product managers need?

Great points from Scott. One perspective I'll add:

"Too many" implies the customer isn't motivated to answer the questions presented. Too many questions might be 1 question, or 15 questions. Therefore this is also a problem of UX design. Some things to consider:

• Help a customer understand the value / impact of what you're asking (why it will help them)
• Ask at the right time! Sending a survey via email is really hit and miss. Maybe try to ask them when they're using your product; this might be via a Typeform modal, or Intercom Messenger etc.
• Use progressive disclosure / logic where possible, so that answers are contextual. You may even consider using a bot, so that the survey seems conversational

Understand that surveys suck! Rarely do I hear customers say how much they love taking surveys. However, they are a necessary requirement to understanding pervasive challenges and needs of your customers.  

I usually follow a few rules of thumb:

1. Have a general hypothesis that you need to validate or have understanding of what you're trying to learn. Perhaps you're trying to understand why customers are churning. Or trying to validate the next set of features. Talk to a few customers to hear their challenges then use surveys to understand pervasiveness.
   
2. Do try to keep them short - unless you're providing them with an incentive, anything more than 10 questions and your inviting fatigue and your responses will reflect that.
   
3. Use logic in your surveys if you need layered questioning - trying to understanding drivers of satisfaction / dissatisfaction with your product, you'll need to ask layered questions to lead you to the aha. Having logic will help you move the respondent through an optimal path for feedback.
   
4. Ask for contact information for follow up. Leverage those that are willing to talk in greater detail about their experiences.

Check out this link for some added insight into survey building - https://www.qualtrics.com/blog/10-tips-for-building-effective-surveys/

Agreed that surveys suck - you're usually not getting near enough responses to constitute a representative sample. Worse, they're often used as a crutch to prop up assumptions. But they're a necessary evil sometimes so to add to your list, here are a few principles I use: 

• Make sure you're prepared to act on the data.
  For each question, ask yourself, what will we do if the answer is A? What if the answer is B? If you don't know, or they aren't different actions, maybe you don't need to ask the question. 
• Don't ask for identifying info, it's a waste of a question.
  You're probably emailing your recipients right? If you sent a survey to their email address, they shouldn't have to tell you what their email address is. With URL variables and hidden fields, most survey tools (Typeform, Survey Monkey, Survey Gizmo) can grab any identifying contact info from your email program. Make sure you have a unique identifier like an email address or user ID and you can then marry responses up to all that great demo/firmographic data in your CRM. 
• Close the loop.
  Tell recipients what you plan to do with the data and then let them know what you did with the data. Customers want to provide feedback but if they feel like they're talking to a brick wall, they'll stop talking.