Underseat storage in exit row
 

All bags go overhead for takeoff and landing.

As long as everything slides completely under the seat in front of you, that's permitted. It can't be sticking out at all.

I see my answer conflicts with the one above. Is that new?

If it is a BA flt. they require bags for EXIT rows to go in overhead compartments.

Which is likely not going to be the case if there is an emergency stop and the bag slides back out!

Same for LATAM.

But the OP is asking about AA 737-800.

Yep.

also, when flying a 32S and I'm in the exit row with no seat in front of me but two up I put my bag under that seat, often to the dismay in the middle seat in the row in front of me. They have their own space in front of them just as I do. Mine is just farther away.

Your personal item, yes. Your rollaboard bag, no. If you're not familiar with it, your personal item has smaller allowed dimensions than your main "bag"; examples of personal item are a laptop bag or a smaller backpack (I have a photo+laptop backpack which can barely hold my 14" laptop as well as a camera and several lenses, and it's right at the limit of the personal item size). And, as mentioned above, your personal item has to fit completely under the seat in front of you, with no protruding straps/etc, in an exit row on AA.

And be aware that the horizontal space under the seat in front of you can vary, depending on which type of 737-800 you have. If you have the 737-800 with in-seat AVOD, then the window and middle seats have entertainment system boxes taking up some of the space. But the aisle seats start with less space, no I'm not sure if you end up with any more net space in the aisle seat than in a window seat with the entertainment system box. But AFAIK there's no way to tell whether you'll have the 737-800 with or without the in-seat AVOD; On the LAX<->BNA nonstop route, for example, it tends to alternate between the two types of 737-800s (which both have the same seating chart AFAIK). Whether it has AVOD or not depends on when AA ordered the plane, no it's a difference of vintage, not model.

Sat in 14A on an -800 yesterday and stowed my backpack under the seat in front with no issues.

If there's an emergency stop, the momentum will make bags slide forward.
(Unless the pilot put the plane in reverse, that is. ;))

This is not correct

And then the overhead at that 32S row is filed with cabin equipment (raft?), makes perfect sense. Actually back at the second set of doors is worse, was back there fortunately only with a backpack, but the lav is directly ahead, so all you have is above, and it's already full before boarding.

The law of physics disagree with you.

On AA, bags may go on floor in exit row but they -must- be completely clear of the walking area. This is in contrast with BA (as others have noted) where you may -not- place anything in the under seat area of the exit row.