You would connect both positive and negative of your fan to the PSU rail, but the positive side to the fan would need the inline fuse.

You can connect it without a fuse, it’s just risky and I discourage doing that.

Those connectors would work for a secondary supply if your laptop charger is 12 volts. Most laptops charge at voltages higher than that though (15 to 20v is typical range) so read the charger spec carefully.