Alice Springs sits on the Todd River at the northern edge of the MacDonnell Ranges, close to halfway between Adelaide and Darwin and about 1,500 km from each. The river is dry most of the year, dry enough to host a boat race, the Henley-on-Todd Regatta, and the town is the usual base for trips out to Uluru-Kata Tjuta.
Camping here means a caravan park in town with the ranges on the doorstep, so the choice comes down to which side of town suits you. Discovery Parks Alice Springs is at Ross, while Alice Springs Tourist Park in Araluen is a short walk from the arts and museum precinct. Both have powered and tent sites, showers and wifi.
Most of what there is to do is close to the parks. The Araluen precinct has the Araluen Arts Centre and the aviation museum side by side, and in the old town centre, Stuart Town Gaol makes a morning of history. Just south at The Gap, the Royal Flying Doctor Service base is open to visitors, and out on the western edge, Alice Springs Desert Park is where to see the desert's animals.
Anzac Hill, right in town, looks across Alice Springs to the ranges, and Heavitree Gap Lookout sits above the gap the Todd River cuts through them. East of town, Emily Gap is a short drive, and about 20 minutes west, Simpsons Gap has a permanent waterhole where black-footed rock wallabies are sometimes seen at dawn or dusk.
Yes. 4 campgrounds here have powered sites confirmed, including Discovery Parks - Alice Springs, Alice Springs Tourist Park and Alice Springs Heritage Caravan Park. For the rest we are still confirming, so check with the park or holiday park before you book a powered site.
Uluru is about 440 km south-west of Alice Springs by road, roughly a five-hour drive, so it is worth planning at least one night at Yulara.
Facts from Wikipedia, "Alice Springs" (CC BY-SA 4.0), Tourism NT, Simpsons Gap and Wikipedia, "Uluru-Kata Tjuta National Park" (CC BY-SA 4.0).

Region boundary: ABS ASGS Edition 3 Tourism Regions 2021 (Tourism Research Australia), CC BY 4.0.