NetBIOS over TCPIP (NBT) is required for "Network Places" (aka The Browse List) to work. It's an ancient mechanism that really hasn't changed any in the past 15 years, but if you disable NBT the browse list disappears with it. NBT is enabled by default.
Your machines are presumably DHCP clients so their gateway settings arrive in the form of one of the options in their DHCP lease from their DHCP server - the router. You shouldn't have to manually fiddle with gateways, and if the machines can ping each other by IP and access the internet, you can be sure their gateway settings are fine.
Re:
1) You don't have to share the port but you might need to feed a specific printer driver to the laptop. If on the desktop you create another printer definition for the same printer, and use the "generic / text-only" driver, then try to connect from the laptop and that works, you can be confident that the problem is the (lack of) real printer driver on the laptop.
2) Probably another firewall effect. If you disable the firewalls (no,
really!) and leave them off for 15 min at least, the browse list should fully populate on both machines, assumbing NBT is also enabled, their Server services are both running, and at least one of them has the Computer Browser service active and running.
3) Try doing this first from an elevated CMD prompt on the desktop (run CMD as admin):
net use
\\laptopName\IPC$ /u:laptopName\accountName accountPassword
It should say "command completed successfully". What happens when you try connect to the laptop's admin shares after that?