Endurance doesn't affect how long it takes to select a grip. Changing your grip after selecting it though can cause you to lose the faceoff.
Yes, I have on-ice trainer on. It says "good timing" in green, but I still lose the Face Off. It will also say "Toss up" but I'll lose 100% of those too. Which is weird because I thought a "toss up" was... a toss up? 50/50. Its not often I run into these matches, but I do come across some players that seem to win no matter what I do. Not sure if they are cheating or whatnot...
We're not aware of a way to cheat on faceoffs. With a toss up, the player with the higher faceoff rating wins. If both have the same FO rating, the winner is randomly chosen.