Ah I didn't see this before I replied to your other thread about Regent vs. Radian.
When installed, the Regent shouldn't move more than an inch at the belt path, and ideally you can get it in there rock solid. What installation method did you use? If you installed it with LATCH, you can try to tighten the LATCH straps up more snugly by reaching through the seatbelt path slots on the Regent's seat back and pulling the LATCH straps toward you to tighten them up more. Sorry if this isn't a very clear description, it would be much easier to demonstrate this in person - a tech showed me this tip when I bought my DD's Regent.
If you can't keep the Regent from moving excessively with a LATCH installation, you could try to install using the seatbelt instead and see if that gets you a more solid installation. In my vehicles (DH's Saturn L100 and my Sienna), installing with the seatbelt using the long belt path method was way more solid than LATCH or the seatbelt using the short belt path method.
I'm not sure about your top tether question (not a tech, just a very carseat obsessed mom), and will defer to a tech to help you out more on that part of your post.