It fixes the paper faster, and then washes out more quickly.

Ilford promotes their non-hardening fixer because of this advantage.  But
Kodak Rapid Fix is just the same if you don't add the hardener.

Note that Kodak paper fixes more slowly than Ilford paper even in the same
fixer, because Kodak's emulsion is also harder.  I was bitten by this a few
years ago... was turning on the lights too early, before the paper finished
fixing, even though my technique had been fine with Ilford paper.

The selling point for hardener was that it guarded the emulsion of the
film/paper against scratches.  But in many years of leaving the hardener
out, I've never had damaged emulsions, even with fairly vigorous
handling.  And hardener does cause the emulsion to retain the fixer,
thus necessitating much longer washes, even with pre-wash treatment in
hypo-clearing agents.  Hardener also makes spotting prints much more
difficult.  The spotting fluid doesn't penetrate the emulsion as quickly
or as evenly as it does on a non-hardened print.  Your method of mixing
seems fine.  Just leave out the hardener.

hardener makes spotting of prints more difficult (spottone will just sit on
surface rather than penetrate) and increases wash time for adequate removal
of fixer (not so much of a problem if hypoclear is used).  it also tends to
make fiber-based paper curl up more which can make it difficult to handle.
i use fixer w hardener on work prints on rc.  finals get no hardener and
extra washing just to be safe.
film always gets hardener.
good luck and happy printing!
-c.

Many people don't bother with the hardener at all, if your careful with
wet film in the darkroom, you really don't need it with modern films.
Hardening does extend the wash times, it also does not work with all
rapid fixers.  Even 20 years ago, I didn't use a hardener, and never had
a problem with damaged film because of it.  I also used most chemicals
one-shot style.  Mix up a batch of fixer, pour whats needed into the
film tank, do a roll and pitch it, a little more expensive, but knowing
every roll had fresh developer and fresh fixer meant that I didn't need
a stop bath, and that results would be consistant.  The oldest film
archive shows that this worked, even though the films are over 20 years
old, no staining or problems due to exhausted chemistries.

With paper, I would put half what I needed into the tray and add water
to get what I needed, process say 10 sheets of paper, then pitch it and
mix a new batch.

W

No. Mix TWO batches. DO NOT use one kind for both. Kodak explicitly
says not to do this. You should not be using powder fixer anyway. I
use the hardener for both film and paper. It does not harm anything.

Ignore the troll. And _don't_ mix up the whole bottle. Mix up
(from the concentrate) only what you will need short term.

Do not use the same for paper as for film, but concentrates
last longer than working solutions...

Also, many modern films don't require hardener, but it never
hurts to add hardener to film fix and _some_ films still need it.
(Tri-X, I think is one according to Kodak.)

Would a post-fix iodide rinse convey the same protection?

 I've read that the amount of silver iodide in IODIZED film
runs two to three percent. Of course non-iodized films have no
silver iodide. From my reading I've found that silver bromide
is by and far the main halide of silver in films.
 Papers, save for one such as Azo which has silver chloride
only, contain silver chloride and or bromide. I think it safe
to say that most projection speed papers contain a mix of
the two. Kentmere claims a little silver iodide in one
of their paper emulsions.

 The ammonium ion will complex with silver in the presence
of iodide but only to a very slight degree. The thiosulfate ion
will complex with silver in the presence of iodide very much
more so.
 All of the silver halides, chloride, bromide, and iodide,
are considered insoluable; the less so in that order. Only the
thiosulfate ion's great affinity for silver in the presence
of any or all of the halides makes possible silver gelatin
photography as we know it.
 If anybody doubts it, fix with fertilizer, ammonium sulfate.
That and ammonium chloride may work well with Azo.        Dan