I’m trying to play on a labtop :
Intel(R) Core™ i5-8300H CPU @ 2.30GHz (8 CPUs), ~2.3GHz
Ram 8gb
Nvidia geforce gtx 1060
ssd
The download is currently at 58% because my internet is very slow. I’m getting horrible performances (stutters/drops every 3 seconds) on minimal settings.
Note that I can’t seem to keep downloading while the game is running, it always pauses.
Can i hope better performances once the download is finished or is this unrelated and i can forget about that labtop ?
A 1060 can run it fine.
I play on max settings on a 1080 at 1440p.
It would seem like a CPU overhead issue.
Once the game is downloaded and the CPU can focus on the game only you should see better performance.
A quad core is the entry level, so with your CPU managing the download and trying to run the game, you will see lag spikes etc as normal.
Your spec can run the game according to what you say you have and what the game requires. 
My gtx 1060 HP omen runs it fine, and I had a similar experience when I wanted to test whilst it was still downloading.
I am genuinely amazed with the performance. I cancelled my 4070ti order after the beta weekends it was that good. 
The thing that concerns me is that the download stops when I run the game and there’s no option to keep downloading in the background, so I’m really not sure if it’s supposed to impact the performances
However, i found out that deactivating pagefiles on my hdd (i have a ssd with the game on, and an hdd) stopped constant 100% usage spikes on the hdd. That seems to dramatically improve everything, for now.