Use South entrance Turkey Flats is a campground. You will have to pay the fees to get into the National Park which is 30.00.

Agree with one of the other reviewers: go up and enter through the main gate near the town of Joshua Tree. We entered through the town of Twentynine Palms but I think the other entrance gives you a better run and sights. Drive all the way through the park and exit at the southern gate but rather than getting straight back on the 10 to head back to your campsite, go under the 10 (or is over) and continue on Box Canyon Road and Printed Canyon Road until the town of Mecca where you can then head back up to the 10 and your campsite. It's a really nice run and different from Joshua Tree Park and it's an extra hour or two well spent. We did it in June and had a great time.

It will be hot, but the heat is dry. We were there in April, and it was already heating up. Hike early in the morning, and, as already noted, bring plenty of water and sunscreen.

You pay only at official park entrances. They open at 8:30am and gates can be closed prior to that. Call to park to check it, but Joshua tree NP has a lot sand/gravel roads that you can use to enter the park in any time. If you managed to cross park boundaries nobody will check if you paid or not. Couple times I returned from park after 7PM using West Entrances and gates were open.

Dogs are allowed in the park on paved and dirt roads, picnic areas, and campgrounds. They are not allowed on any of the hiking trails or buildings. Rule of thumb, your dog can go anywhere your car can go.

take southern entrance

No idea, maybe you should ring visitor centre or tourist office?