Is it allowed to take a safari with an own car in South Luangwa and other national parks?

Yes, you can take your own vehicle into any Zambian national park. You can't camp where you like - you will have to stay at a lodge or similar - but you can drive yourself around all of them on your own.
You certainly can - just turn up at a gate and pay your money.
If you are going to a lodge, you are probably automatically be charged for game drives in their vehicles. Also research where they are - some (eg Kafunta) are quite a drive from the Mfuwe Bridge gate - their vehicles have access to a pontoon/ferry not far away but private vehicles aren't allowed to use it.
There are a couple of campsites close to the Mfuwe Bridge - Flatdogs being one of the most well known (but check as I think I heard that they didn't allow camping during some parts of the year) but there are other in the same area.

Thank you! That's good to know. Now I only need to find enough money and time to travel to Zambia :)

Hallo,
ew did it in september and we stayed at Wildlife Camp. We visited the other camps, but Wildlife looked as nicest.
Beate
We chose to not rent a car because they are expensive in Zambia.
Instead, we road buses and took private taxies.
Flatdogs can provide you with info on how to do this. We loved Flatdogs and stayed there for over a month. Saw elephants and other wildlife in camp, took their game drives into the park about once a week. Check their calendar because they indeed only allow camping part of the year.
South African tour leader we met at Flatdogs takes his groups to Wildlife Camp when Flatdogs is closed to campers.