Yes, the moment.js library is used in many parts of the front-end (public calendar, cookies alert, list of plans, statistics, shopping cart and coupons). It fetches the date from the client browser (which relies on the client operating system date).
So it could be responsible of some problems if it is not correctly configured. Especially, your first screenshot display an error in the "shopping cart" component, so it may be the cause of the issue.
Concerning your second screenshot, the admin calendar is not using moment.js but directly the native browser date in some parts on the code so maybe it's linked too...
Do you have a way to check if the issue is linked to that?
Concerning your other question, I'll answer here, but please yes: a separate thread would be better for the next time
Starting with fab-manager v4.3, the next version to be released in some days, you'll be able to configure a custom duration for reservations slots, using the
SLOT_DURATION environment variable. I'll keep you updated when it will be released.