Bring all who claim me as their God, for I have made them for my glory. It was I who created them.'" Isaiah 43:7 NLT
for through him God created everything in the heavenly realms and on earth. He made the things we can see and the things we can't see--such as thrones, kingdoms, rulers, and authorities in the unseen world. Everything was created through him and for him. 17 He existed before anything else, and he holds all creation together. Colossians 1:16-17 NLT
For we are God's masterpiece. He has created us anew in Christ Jesus, so we can do the good things he planned for us long ago. Ephesians 2:10 NLT
In trying to answer your question at the end. I believe we were created to have a relationship with Him which you allude to. I think about God (Gen 3:8) walking in the Garden, where he had placed man. However, I also believe that in making His creation, He also wanted this relationship to be voluntary. So, for it to be so, man would have needed the ability to make choices, in a way that they could accept or reject Him and so conditions were made that allowed man to do so. Which you also allude to
I do not see it as a "project" or "experiment," because an assumption would imply that God is not Omniscient or in other words, all-knowing. How could it be an experiment if He knows the outcome unless He limited a part of Himself from this knowledge in the experiment instance that you mentioned?
However, I do see a project as a possibility but when considered in light of the purpose which I believe is for His Glory and is acceptable, while still being omniscient (fully knowing the results thereof).