The Roots of Psychology are in the human inner motivation to learn. In this sense, Aristotle would affirm that "all men by nature desire to know". I would add the sense of mystery. All men desire to know the roots of mysteries and humanity is a mystery for humanity. This (mystery) is good for human race. Do not look for our roots outside us, look for them under us (or inside us), just like a tree would do if it were looking for its roots. (This is stated by many HIstory of Psychology books I have read).
Psychologists in many different context and situations have used different process or methods in order to get to the answers they need. From Greek psychology to Wilhelm Wundt, the criteria differ vastly. Introspection, observation, statistical analysis, questions and answers, psychotherapy, and so on. I highly recommend Robert Feldman's book "Understanding Psychology" in order to get answers to your last question.
There are different perspectives on the history of psychology,. Some of them go back to more than 2-wtothousand years ago, others start in the Renaissance and others, like you, start with Wundt. when psychology becames an academic discipline. The latter is situated in the beginning of behavioral 'sciences', like economics, sociology, psychology.
Interesting information that covers a wide range of interpretations is found in following exemplary books:
1. Older but relevant set of different contributions
Basic Writings in the History of Psychology. Robert Irving Watson (1979) Oxford University Press, USA ISBN 0195024443
2. Mainly focused on ‘modern’ psychology A brief history of modern psychology. Benjamin Jr., Ludy T. (2007) Malden: Blackwell Publishing
3. Large historical overview. Psychology: Pythagoras to Present John C. Malone (2009) MIT Press (MA) ISBN 0262012960 (ISBN13: 9780262012966)
4. Western psychology from Renaissance on. From Soul to Mind: The Emergence of Psychology, from Erasmus Darwin to William James. Edward S. Reed (1997) Yale University Press ISBN 0300069677
I think it is very close interaction between mental processes and behavior. If you study 'praxis' you may get another variable in this regard. I hope it helps!
One answer to this question comes from examining the roots of the word "Psychology." You'll find a lot of references in the following articles which are available on ResearchGate
Origin and evolution of the term Psychology: Addenda.
Hendrika Vande Kemp
Jul 1980 · American Psychologist
A note on the term “psychology” in english titles: Predecessors of Rauch
Hendrika Vande Kemp
May 1983 · Journal of the History of the Behavioral Sciences
The tension between psychology and theology: The etymological roots.
Psychology owes much to empiricist philosophers such as David Hume. Noam Chomsky's more nativist ideas about language acquisition owe something to the rationalist Descartes.