The High peak trails are a drive but can be done on your bikes, parking is especially tight/limited so taking your bikes is an advantage. You will see that the rangers will block the road to the peak parking when it is full. But there are also flat hiking trails right from the campground too that can be explored. Absolutely zero internet/phone data services. One pay phone by the store.

Yes, there is a shuttle between the campground area and the high peaks trails. You can also ride your bikes.

Your bikes will be helpful - it's over 2 miles from the campground to the closest trailhead. It's doable by walking but adds a fair amount of mileage. The road from the campground to the trailheads has little traffic and should be bike-friendly.

Yes, absolutely. We never used our car while we were there and hiked all over including the bat caves.

You can hike to all of the trail heads except the High Peaks Trail on the west side of the park, which is accessible from other trail starting near the campgrounds, but hiking to them from the campgrounds adds several miles to the hike. There is a lovely trail that leads from the campgrounds to the staging area for the most popular hikes in the east side such as the Bear Gulch caves trail that adds two miles one way to whatever hike you are taking from there. You can bike to all the eastern side trail heads though there are no bike racks on at the trailhead for the Old Pinnacles Trail. Depending on how far you want to hike all areas are accessible from the east side. Log on to the NPS site for that park and there are good maps with distances and descriptions of the hikes.

Biking to the trailhead, especially on a weekend, is a great way to go. There are bike racks at each trailhead. We were at Pinnacles in November of 2020 and the shuttles from the campground were not running because of COVID.

While you can access one trail directly from the campground, most (the caves and the Condor Gulch Trail) are a bit further away as I recall. I would think you could bike to some of the trailheads.

We didn't have our bikes, but wished we had, in class A. They had closed the road to the trails (for all vehicles) at the campground and were open to bikes only. You may want to call the main office and chat with them. We didn't bring the bikes due to the roads being narrow. Also, the electricity at our campsite and others, came up on our surge protector as "ungrounded", so beware.