Better do that:

if [ -d /var/www ] ; then touch folder_exists ; else touch folder_absent ; fi

The first version of the command does the following: $( ... ) is first replaced by whatever was produced by .... Since you call ls -l | grep www, you are either getting a line with multiple words, or nothing. In the first case it is attempted to be executed, but that makes little sense. What will you get if you type

drwxr-xr-x 15 root root     4096 Aug 20 13:48 log

at the command prompt? There is no command called drwxr-xr-x. If you use the cat $( ls -l /var | grep www) form, then first the $( ... ) is replaced by what was produced by ..., then cat interpretes it as a list of files to cat (starting with drwxr-xr-x; most likely, there is no such file).

Moreover, if you attempt to do something like that:

command > cat file

What will happen is that cat is interpreted as a filename to store the output of command, and not as a command. To interpret cat as a command, you need a pipe sign, like you correctly used in ls | grep:

command | cat

However, if your purpose is to store the output of command in a file, then just use

command > file

As a matter of fact, you could do

ls /var | grep www > file

This file will be empty only if there is no www folder in ls. But beware! if there is a folder /var/ewwwocks, then the file will also contain a line. Better to use simply

ls /var/www > file

Long story short, use the conditional construct if ... ; then ... ; else ... ; fi