I feel like I should refuse to bring the carts in at my job. They ask me to keep the women from having to go out in the rain and do it.
Are they really protecting the women from getting wet?
Yes, I want them to work more than me.
While you are right pointing out the sexism and judgment, you need to still follow your orders and follow your boss at this time. You signed up for this job so you need to commit first, complain later in your off hours. You risk ruining business with your complaints.
I'm frustrated that I have to keep the carts and I don't want to get muddy. I'm going to file a complaint with HR.
If your job is the cart pusher I'm not sure HR will do anything about your complaint. If your boss tells you to do something work related you should just do it or start looking for another job.