'Respond ' could also mean to react ,this reaction could be a verbal reply or just an action .
He responded to her email quickly by calling her on the phone .
He responded to her email quickly by only 2 words " me too " .
in both cases you do examine , inquire , try to find out about something in a systematic manner , yet 'to investigate ' is more concerned with allegations, accusations, detection of crimes etc ...
The research is more associated with scientific issues .
In fact they are still used in both fields interchangeably
He kept researching the close relationship between the 2 phenomenas for more than 2 years.
He investigated the case to decide ,whether there is enough evidence for the crime or not .
inquire /look into
They could mean the same.
To inquire is also to ask for information about something from a certain source, it also means to look into and investigate .
I hope this helps a bit. In fact as much as those verbs do intersect at a point ,where they have the same meaning , as much as they could have different meanings and won't be interchangeable in certain contexts .