I am confused by the first issue you’re having. I never had this reported or came across it myself (also failed to reproduce it just now).
Also when you say “browser”, do you mean an actual Browser or the Syncthing UI embedded window in QST?
Do you launch Syncthing yourself or through QSyncthingTray?
Can you add some more information? Like macOS version and Syncthing version?
Regarding the energy, try increasing the “polling interval” a little.
There once was a bug with significant energy caused by the dedicated graphics card being used, but this was fixed a few versions ago.
Can you check whether you have “restartOnWakeup” checked in the advanced settings?
This causes Syncthing to restart when resuming from standby (thus opening the browser).
Since your mac is properly connected to power and therefore goes in and out of standby several times a day, this might cause the browser windows (QST itself does not open any Browser except its embedded one).
The polling interval is in seconds. Good point, will add that to the UI.