Games belong to PSN account and not to specific PS4 device.
Once you login to your new PS4 with your old PSN account, go to Library and you will be able to download your games (see Download PlayStation Store Purchases to the PS4).
If you want to avoid downloads, you can bring your games via external HDD or flash drive (see Can I copy PS4 games to USB stick (...
You cannot transfer PS4 games between accounts. If you want the games split between two separate account, you will have to buy the games again unfortunately.
I found this on answers.microsoft.com: (thank you Kapil Arya)
The below worked instantly and error free for me:
*To use your PS4 controller as a game controller, you would need to download a driver for it.
Download the driver on this link: https://github.com/Jays2Kings/DS4Windows/releases/download/v1.4.52/DS4Windows.zip
Extract the content of the compressed ...
Yeah, even reddit complains about a lack of rebinding. Unless you alter the layout you're using, you can't change the controls.
The best you can do is attempt to remap your controller, but beware that that changes your controller mapping for everything and is generally pretty unwieldy.
Minecraft on PS3 and PS4 are two different games, so while you may own the PS3 version, you will have to pay again to get the PS4 version. Only games that support "Cross-Buy" can be bought once and played on different systems, but you will have to read up on what the game supports on the PlayStation Store.
Cross Buy games will usually have tags for ...
I don't know about Minecraft specifically, but the only PlayStation games that would allow you to do this would say in the game's description that you get both versions when purchased. Most such titles are for console and portable versions, I'm not aware of any PS3/PS4 ones. So, either buy the PS4 version of Minecraft or hook your PS3 back up.