Do I need to open my eyes for Face ID to work?

1. No. Eyes must be opened and you must be looking at the iPhone for Face ID to work.
2. No. You can use one or the other, both are not required to unlock. The passcode may be required instead of the Face ID option under the following circumstances:

To use Face ID, you must set up a passcode on your device. You must enter your passcode for additional security validation when:

• The device has just been turned on or restarted.
• The device hasn’t been unlocked for more than 48 hours.
• The passcode hasn’t been used to unlock the device in the last six and a half days and Face ID hasn't unlocked the device in the last 4 hours.
• The device has received a remote lock command.
• After five unsuccessful attempts to match a face.
• After initiating power off / Emergency SOS by pressing and holding either volume button and the side button simultaneously for 2 seconds.

click here ➜ About Face ID advanced technology - Apple Support

1. Yes and No. Normally, you cannot unlock your phone with your eyes closed. I DO NOT RECOMMEND DOING THIS, but in Settings > Accessibility > Face ID and Attention and turn "Require attention for Face ID" off. Do not, I repeat, DO NOT do this. This will cause someone to be able to unlock your iPhone no matter what and puts your security at risk.
2. Refer to Phil0124's reply.