just my opinion, but the connection you mention in the first sentence above is hugely important, huge! As it relates to what you said in the second in regards to using your body that way when you do aikido, you should be working to feel that sensation or connection in everything you do, before, during and after. IMO, the whole second nature stuff will never happen until you start feeling those things outside of solo exercises doing every day life stuff. So work what you have, what you can feel and just continue to build.
Yes, I think these sensations come from the connective tissues targeted by this training. Strangely though, some of the sensations don't feel like connective tissue to me. It feels more like coming from the skin itself. Sometimes I do feel that outside of training, but it's mostly accidental. I haven't really spent a lot of attention to it. I'm not aware that these sensations have a function, besides letting me know something is working different than before.
I like your tip. I'll pay more attention to getting these sensations outside of my training. Thanks!