With or without the trainer, that red circle appears if you try to move a unit too soon
As long as I don't go crazy with the unlimited movement, there's no problems with that. I've still not seen the "Remove unit damage" cheat work yet. Super Resources seems to work just fine. Golden Age points, faster building creation, faster research, increased citizen creation, and money all appear to work without any hiccups. Oh, and max unit XP works also.
I've never really used any of the others much.
firewall and other security software.
on trainer and game.
version and distribution.