Winrar doesn't run unless you want it to (open it) and doesn't require an internet connection. Why would that affect your computer? I've been using it for years and never had a problem. I like it better than Winzip because it's free (if I recall correctly and things might have changed over the years... if you end your trial on Winzip you need to pay. Winrar you're able to continue past the trial without upgrading and have no limitations).
Oh and Winzip does open Rar files. Looks like it supports:
7Z, IMG, ISO RAR, BZ2, CAB, LHA and other archive files
Winrar supports these files:
RAR, ZIP, CAB, ARJ, LZH, ACE, TAR, GZip, UUE, ISO, BZIP2, Z and 7-Zip