Remember that parameter expansion doesn’t happen magically, you need a shell to do it for you.

 var=3 /bin/echo `$var`

I’m not sure what you’re trying to do with that one, back-ticks aren’t appropriate here (they execute a command and are replaced with its output).

 var=3 bash -c "/bin/echo '$var'"

doesn’t work because the child shell is run with

 /bin/echo 

and single quotes doesn't prevent parameter expansion in the current shell. Try

 var=3 bash -c 'echo $var'

instead: the single quotes here protect the argument from the current shell, and the new shell is run with

 echo $var

which it will process as you’d expect.

In your second point, my initial comment still applies. If you want to play around with the environment and see what happens, run `env` and filter its output instead:

 var=3 env | grep var
 var=3 exec env | grep var