If I download games from the PlayStation store on my PlayStation 4, and then my PlayStation 4 somehow broke and I bought another one, Would I still have those games I had on my old one, on my new one? Or would I lose those games?
It depends on your absolute definition of "lose those games", but they would certainly not be lost.
When you purchase a game on the PlayStation store, you are purchasing it to your account. Providing you are logged in to the same account, you are free to go back on the PlayStation store, and download the game again for no additional charge.
So in regards to the brand new PlayStation 4, you would still have to download the games. Assuming you are logged in to the same PlayStation account, you will not have to pay for them, again.