It seems that there is definitely a problem with the LCD screen on the phone and probably more to the systemboard as well judging from the picture and your description.
Here is a link to the Ifixit guides on how to repair the phone.
Samsung Galaxy S5 Repair
It may be that a screen replacement will fix at least the obvious damage to it but whether it solves everything would have to be determined. I somehow doubt it.
I recommend that you take it to 3 reputable, professional mobile phone repair services and ask for a quote for a repair. I suggest 3, that way if all the repair estimates are close you will have a truer indication of the cost of the repair.It may be that they say that it is not economically viable or even technically possible to repair the phone.
If they can repair it (or say they can) then it becomes an economic decision for you to make.
Hopefully this is of some help..