They 2019 HSA contribution limit for a single person is $3,500. If my employer contributes $500, does that mean I can only contribute $3,000, or can I still contribute the full $3,500?
Yes, the contribution limit includes anything contributed to your HSA, whether added by you or your employer.
From IRS Publication 969:
Employer contributions. You must reduce the amount you, or any other person, can contribute to your HSA by the amount of any contributions made by your employer that are excludable from your income. This includes amounts contributed to your account by your employer through a cafeteria plan.
If your contribution is $3500, and your employer puts in $500, then you can only put in $3000.
Yes, employer contributions do count.