Is there an entrance fee, and is cycling permitted?

There are trail heads into the park near residential areas. Although some may have been closed do to the amount of traffic through the area. There are some where you can park on the street near the trail entrance and ride into the park. Mountain biking is permitted. Fees are $5.00 per vehicle ($4.00 seniors age 62+). Pay cash, check, or credit card to kiosk attendant or at the self-pay Iron Ranger box next to the kiosk. Camping: Chino Hills Campground is open 7 days a week $30 per night 8 people maximum per site. October - March 8am-5pm 7 Days a Week April - September 8am-7pm 7 Days a Week

Yes there is entry fee. And they allow cycling there.

When I self pay with the box, how do they know which car payed for the parking?

Yes and yes..... And there is a parking area.... It's worth it!