I have a device without public IP, AFAIK behind NAT, and a server. If I use bore to open a port through my server and host a game, and my friends connect to me via IP, will we have big ping (as in, do packets travel to the server first, then to me) or low ping (as in, do packets travel straight to me)?
No, you will have “big ping”. bore (and everything on that page i linked) is strictly for tunneling which means all packets are going through the tunnel server.
Instead of tunneling, you can try various forms of hole punching for NAT traversal which, depending on the NAT implementation, will work sometimes to have a direct connection between users. You can use something like tailscale (and if you want to run your own server, headscale) which will try its best to punch a hole for a p2p connection and will only fall back to relaying through a server if absolutely necessary.