Re: Moving with your center
the reason why it's very difficult to understand this concept is because there are prerequisites before you can accomplish this consistently. you need to learn first or discover how nage and uke are connected and how the connection is manipulated. Moving with the center is just one of the many principles in effectively manipulating the connection.
although i wont be able to give you answers to your original question, what i suggest is that when you train, that you not only focus on what you train but how you train.
Aikido is a difficult art to learn and it will take years/decades to master. Only by constantly practicing will you get the answers. the answers though will present themselves one at a time. It may take a very long time to get an answer to some of your questions but if you focus on how you train, you can accelerate your pace of learning.
What helped me accelerate my learning was that I told myself to be fully aware of what's going on. observe sensei carefully, observe ukes reaction during a technique, observe yourself when moving. In aikido, the answers are in the smallest/finest of details. They are very subtle but significantly helps with techniques. next, you will then realize that there are commonalities (principle/moves) regardless of what technique you are doing. look for these common things. by being aware of the smallest details will you have epiphanies coming to you more often. One by one you will "get it".
sorry i didnt give you direct answers but hope that the above is of some help to help your aikido.