Having a faulty or weak Alternator, would that have an effect on the automatic transmission shifting gear takes place early. Moreover the kick down doesn't work?

Faulty alternator and low battery voltage can cause strange things to happen. No way to know if that is the problem with your transmission or not. Just replace the alternator and see what happens.

Thanks a lot for your answer. I replaced the alternator but still same problems exist. Auto transmission changes early and kick down does not work. I will take it for a computer test.