Consider used? I like to Google best gaming laptop lists by year and then cross check any system tests and the current prices on ebay. I skip Alienware and other known "gaming" systems since they still command a premium. I'm shopping in the $200-$500 range, but can find a great deal every week or so, when I'm in the market.
I have two Razer 15 laptops with i7-8750 processors, 32gb of RAM and 1070GTX video cards. Each is capable of smoothly running 24 toons as long as I am using eqwire with Turbo DISABLED on the laptop. Comments below:
1 - either 4 or 6 is going to be fine. With a 6 core processor you can disable turbo and run really cool. If you don't you will most likely have a very hot laptop on your hands. The 6 core CPUs run HOT AS HELL.
2 - kind of irrelevant as EQ barely touches the GPU. Any 10xx GPU will likely be overkill
3 - Even though EQ doesn't use the GPU much, I've never had a good experience running EQ with integrated graphics. I think these still share system RAM? That could be why.
Biggest thing is to make sure you have a full power CPU, not a U, and that the laptop is capable of cooling itself properly. Overheating/throttling will ruin your day. Also - EQWire is a godsend for laptops.