Is there any website where I can play simultaneous chess against many opponents? This is purely for my fun (and also practice!), but is there any place online where I can play simultaneous chess against many opponents? 

This would be against human players, of course, not against computers.
You could do that on FICS, here's how:

Simuls

  "Simuls" = simultaneous chess matches; playing more than one opponent at a
time.  It is possible to play more than one opponent simultaneously on the
chess server.  Below are sections for Starting a Simul and for Joining a
Simul.

  Users using an interface that only supports a single board (that is if
two games are being observed a move from the second game will cause the
position of the first game to disappear  (such as Winboard and Xboard)) should
issue "iset singleboard 1" before starting a simul.  This will ensure that
when simul participants move, only the board the simul holder is at will
update.


STARTING A SIMUL

  In order to play more than one chess match simultaneously you need to do the
following:

1. Start the simul with the "simstart" command.  Announce your simul
   conditions on shout and cshout enough to get the word out, but not so much
   as to annoy users.

2. Other users must then challenge you with "simmatch".

3. Accept or reject the challenges as they come in.  The first game will be
   colored as is if was an ordinary match.  The next game will be the
   opposite color, the next the opposite to that and so on.

4. When you are satisfied that you have started the number of games that you
   can handle, turn off your open flag with "set open 0".  You can see your
   games with "simgames".

5. You can then start making moves on the board.  The "simnext" command moves
   you to the next board; "simprev" moves you to the previous board.  You
   will focus on one game at a time, going in a particular order.  If you
   wish to skip to a board of particular interest "goboard &lt;player&gt;" or
   "goboard &lt;game number&gt;" will take you to that board.

6. If you are bothered by how long a player is taking to make the next move,
   or they are being disruptive you can abort that game with "simabort".  You
   can also force an adjournment with "simadjourn".  Only games adjourned
   during the simul can be resumed.  After the simul any outstanding simul
   games may be resumed as normal matches.  Should there be a need to
   terminate the simul "simallabort" and "simalladjourn" will abort or adjourn
   all games respectively.  Both commands will also terminate the simul.

7. When you are finished type "simend".  This will return you to normal
   playing mode and inform you of your statistics for the simul.

Various commands you can use while running a simul include:  goboard,
simabort, simadjourn, simallabort, simalladjourn, simnext, simprev and simend.


JOINING A SIMUL

  In order to play against a user who is open for playing simultaneous
matches, here is what you do:

1. Using the "simmatch" command, challenge the person you will be playing.

2. You will be informed when the simul player is coming to your board.  You
   should be prepared to move either before or shortly after that player
   arrives at your board.  If you do not move within a reasonable time
   period, your game could be aborted.

3. It is impolite to offer a draw unless that player is at your board.  It
   could very easily confuse the simul player if offers come in from
   different users at once.


OBSERVING A SIMUL

  You can observe all games in a simul once they have started by typing
"simobserve simul_player".  All of the simul_player's simul games will be
added to your observation list.  As the simul_player moves from board to
board, so will you.  When the last game has ended, you will see the final
results from the simul games.  You can also use the "follow" command for this
purpose.  Using the "observe" command, however, is not reliable for this
purpose.


Reference.

I don't know if it affects your rating, never tried it. Be careful what you wish for :)

There's also RedHotPawn, it says that  it offers that, maybe a member can verify that, I signed up there a year ago never used that site. But it's the second most popular online chess website after Chess.com I believe.

PlayChess server offers that as well but it's not free, but it's well known server, best of the best.

On Chess.com, you could play live chess and duplicate tabs, and you'll end up playing more than games at once, not the same window though.

There's also gameknot.com, I have never heard of it, but they claim it is possible to play such games on their site.