The cartridge reader is probably bad. It either has bent pins or it could have been returned from someone else with possible liquid damage or something. If you just bought it, I would return it asap and just get a new one. The cart reader is a pretty easy fix as long as your soldering skills are decent but why go through the trouble when you can just return it.
Start by unplugging it, leave it unplugged then push the power button to discharge the power supply. Plug it back in, when you attempt to power it on does it power on, shut off, then not let you turn on again? Or does it turn on, then let you try again and just shut off again?
They don't connect at all? Is it intermittent? Either way, the wifi antenna probably needs to be replaced. I know when this thing malfunctions controllers drop connection and wont sync as well. That being said, your controllers probably wouldn't sync if that's the case so before you attempt this repair I would check other headphones first to make sure. Here's the tutorial for the wifi antenna PlayStation 4 Wi-Fi アンテナの交換 Hope this helps!
You can separate the digitizer from the lcd with a thin flat pry tool. The adhesive is not that strong and you don't need heat but you can use a little if it makes it easier for you. You don't even have to reapply any adhesive if you don't want to given the fact that the tray that holds the main board and the main board both push down on the screen and dig to hold it in place, the adhesive seemed more like a precaution on Nintendos part. 2 separate parts = cheaper repair in your case. Hope this helps!