Yes, a newer Roku should work with any tv with HDMI inputs.
Note that your old Roku may have been one of the old models that drew its power by being connected to an MHL-enabled HDMI port found on some tv's in the early 2010's. MHL never really caught on, though, and is rare or non-existant on current tv models so this capability has been dropped from Rokus in recent years. Now Rokus are powered via separate USB cords plugged into a suitable USB port on the tv or into a supplied power adaptor plugged into household current.
Can't answer the question re the old remote, but the new Roku will come with its own remote.