Wait you wouldn’t take your cat? After a week of you rarely being available you consider not being home at all for two days and a night?
I would strongly recommend against that. I’m worried that this may cause behavioral issues with the cat getting bored, depressed, feeling caged…
Possible consequences could be urinating outside the litterbox, willfully scratching your furniture/walls and more.
I’m not sure if with your work and travel habits a pet would be a good choice. Maybe one that doesn’t need social interaction?
Sticks may be something that he has been hit with before - or maybe it’s the position above his head of something that scares him. You never know what a former stray experienced.
In my experience toy type matters a lot too. My current cat goes crazy over feathers at a rope that you can whirl around with a stick - I guess the sound the feathers make trigger her hunting instincts? Our other cat gets scared of that but hes always been sensitive to noise.
He loves those little snakes that are attached to sticks with a rope though. Leaves the feather cat completely cold. Catnip was meh for ours, but both lost their shit with Valerian, totally drugged up for five minutes (then the effect wears off and the cats won’t feel the same even with fresh scent). Where I live it gets used to help people sleep so we buy a whole bunch at a pharmacy, fill it into an airtight container (that shit stinks, it could make me vomit) and marinate cloth toys in their that we give them every now and then.
At the end of the day, all cats are different and trial and error is best. Also, if you found a cheap toy that works, stock up - often it’s not easy to get replacements once they break.