The old Trinity 4500s had an internal cutoff... but it was kinda of freaky and if used in a ESC that has lipo cutoff it would mess things up. It would also cut off it you pulled to many amps out of it at one time to. Used to happen to guys at our track running 4wd mod.
You can charge your LiPos right after you race.... they should be mildly warm.. but not hot. I charge mine right after... no prob.