First, the easy part. A Lightroom license entitles you to two installations for your personal use. So it won't be necessary to uninstall from your one computer in order to install on the new one. You can have it installed on both and be in full compliance with the license.
Having access to all of your images and your current catalog requires some additional consideration. There isn't a way for you to access the catalog and images on the other computer. All images need to be on a drive that is local to the computer you are using. You could try moving your catalog folder and all your images to a dropbox folder for the time that you are gone. Another option would be to put all of your images on an external hard drive, and then copy your catalog you are using now to the new computer. But using an external hard drive sort of ruins the portability to a certain extent. The other alternative would be to copy your catalog folder and all of your images to the hard drive of the new computer. Then, when you return home, you could replace everything on your home computer with the image files and catalog from the MacBook.