Is it ok to screw those red utility hooks into wall studs, or should I use a crossbar between two studs?
The hook is going to be inserted horizontally, so the load is mostly shear (downward) on the hook itself. You are in effect compressing the shaft of the hook downward on the bottom of the screw hole.  There is some tension pull outward on the wood, but it is minimal for the types of items you listed. Studs made of 2x lumber can easily handle light loads in this configuration.

I would not try to hang items weighing several hundred pounds in this manner.

Also, I would not drill several holes, closely spaced, one above the other. While this would still probably hold your loads with little difficulty, it might compromise the integrity of the stud overall.