I would never ever cross reins or move them over the neck. It is an extremely bad habit, normaly resulting in a crooked horse who falls out through his outside shoulder. It also affects your balance as a rider and the consitancy of the contact.
I've only used spurs on one of my horses, and it was more of a refinement thing. I had trained him to be pretty sensitive off the leg so I in no way needed them in way of general riding but I found them helpful especially in showjumping where I could just angle my foot to give a stronger aid if needed. I guess in the end I used to use them when he'd lean in, but I could do the same thing without spurs but with a bit more work on my behalf. Even when I used them I certainly wouldn't use them every ride, just sometimes, mixing things up. I really don't like it when someone "needs" spurs, or "needs" anything for that matter. I've met people whose horses won't do halt to canter transitions or whatever without spurs, but things take work and people are always looking for short cuts. To get that horse so good off the leg I worked on it everyday, transitions and more transitions, making leg circles etc, and he was super great in the end. I think spurs should be reserved for refinement or for when things are already learned.
I guess listen to your trainer, but I wouldn't want to have a horse that could only do correct 20m circles with spurs. I kind of think they are good for making things easier when you've already got it down pat.