sunshop cart update, customization and whole website with databases transfer to new vps with design updates

My current website is located on an older VPS, where there is an outdated php MySQL shopping cart. The new version has already been installed on a new VPS with completely updated environment, Linux, Centos, php. The rest of html pages and blog have been moved but we continue to update content daily, so the new content has to be moved also. This project involves moving latest additions to databases over including customer databases to new VPS in a timely way so as to minimize downtime, and everything verified as working before we go live with updated site. Had this started by two people, but neither completed it - just did certain parts without big picture. So now I am starting again. Now I need the theme changed to responsive (as that wasn't done) while preserving customizations of appearance, checkout process, and shipping integration. This may also involve some changes to our design to take advantage of the new theme. But I do not want a wordpress site built to replace this, so please don't suggest that. I do need to finish the update soon. As a small business owner, I want to work with someone who will look it all over, take some thought and make suggestions of what has to be done to have a successful outcome, and will communicate with me in a timely way about what I need, what needs to be done to accomplish that, and when it can be done.