Phone screen goes crazy after replacement

@billtonik The first thing I would do in a case like this, is to open the phone up again.
Then I would disconnect the battery (always starts by disconnecting the battery, as leaving the battery connected while disconnecting other cables can do serious damage).

Once the battery is disconnected, disconnect the replacement screen and inspect cables and connectors on both screen and logic board. If everything checks out and you can't see anything obvious (it can be hard to visually tell without a microscope), reconnect the replacement screen making absolutely sure that it's well connected and plugged in correctly.
Reconnect battery and turn the phone on (but don't close it up fully just yet). Inspect it's functionality and "stress test" it a bit to see if the problem persists.

I have experienced stuff like this when a connector was only partially connected, even though it appeared to be well seated.
In that case a tiny lint piece had blocked a pin on one connector, and that created havoc.

If the problem doesn't go away, it's possible the replacement is faulty.