Naturally this depends primarily on how far away from home you happen to be.
If you take a base case of ~1gbps simultaneous fiber link at your house and you are within 10s of miles, the added latency will probably be noticable but not horrendous (back of the napkin says 20-100ms, mostly due to 5g wireless hiccups and cross-network carrier transit).
Browsing HN: May not notice
Watching YouTube or other more bandwidth intensive activities: Probably could notice a little lag and/or longer load times. Even if you have 1gbps upload at home, in some (or many) cases you may only achieve 100mbps or possibly even less to your device (I've tested this manner of Wireguard PTP extensively IRL).
No it just tunnels your DNS traffic and traffic to the 10.6.0.* IPs. Could also add with a comma 192.168.1.0/24 for other local IPs you want to route through WG.