Welcome to Sway club! 😜 This sounds like you haven't installed an xdg desktop portal (https://wiki.archlinux.org/title/XDG_Desktop_Portal) I personally installed the gnome ones. However you can pick any of these. They provide stuff like file pickers, etc. Regarding the links there was a command to set the default app for different mime types using xdg-open. Unfortunately I wasn't able to find it.

The commenter talking about xdg desktop portal is correct for the file picker; you need to install a portal for that.

I can’t get links to open from discord

Are you using dbus? Display managers normally will launch graphical sessions with dbus for you, but if you're manually launching from the tty, use dbus-run-session sway (or the name of the executable you want to run). dbus is used for applications to communicate with each other, e.g. Discord to communicate with your browser. And you need dbus installed and the daemon running, of course.

In case you don't know, Sway is not a full desktop environment like KDE or Gnome, it's kind of a DIY thing. Sway is just a Window Manager, then you need to set environmental variables, the XDG portal that someone already posted, file manager and probably a way to manage themes.