I'm answering my own question based on this question's activity, my own research, and discussions with my financial advisor. 0xFEE1DEAD's answer goes a long way, but it contains a lot of information that isn't relevant to an employee.
You can't know for sure if you are a Highly Compensated Employee. It's a company-specific thing rather than an IRS blanket rule. But it doesn't really matter, because this distinction isn't meant for you. It's a way to help your employer manage their benefit plans to make sure the plans don't favor employees that have higher income. Make your elections as you see fit and as your employer allows.
For your 401(k), it's your employer's responsibility to make sure they comply with the nondiscrimination rules for HCEs. If you are an HCE and they fall afoul of the rules, you may get some money returned to you at the end of the year, but that shouldn't stop you from contributing up to the maximum allowed amount if you choose to do so.
For your Health and Dependent Care FSAs, restricting contributions for HCEs seems to be a company-specific rule rather than a hard-and-fast rule set by the IRS. You can make your elections as your benefit enrollment system allows you to, and let your company limit your contributions if necessary. Hopefully, your company has set up their benefits system so that you can't elect to contribute more than you should be allowed to.
Even if your income is above the threshold to be considered an HCE, it's up to your employer whether or not they choose to classify only the top 20% of employees by compensation as HCEs, based on how they want to manage the fairness of their plan offerings. They don't generally share this information with you. But if you have questions on how your company handles this, you can reach out to your company's benefits and/or payroll administrator.