If there are missing pieces on he logic board you should be able to get it fixed. I’m not particularly familiar with this phone, but as long as it’s not like Apple where you need to tune it, it should work.
One of the main reasons why the screen turns green on this device is due to the battery draining quickly. As you have installed a new battery, there could be two reasons for this fault, stemming from this. The first, would be that the battery you purchased is faulty and will not hold its charge, and the second could be that it had not been calibrated properly. Try calibrating it, and let me know if it still doesn’t work and we can try something else.