Protecting items (particularly a handy haversack) from destruction

[MaxiDuRaritry]

a glove of storing is extradimensional storage. a haversack is extradimensional storage. i'm pretty sure you can't put two extradimensional storages one inside the other.

Nope. Only 2 such interactions occur, and both involve a bag of holding and a portable hole. A glove of storing is neither.

Unless we're houseruling, and you don't even need an extradimensional space for that.

I wouldn't say that.
sure, if you drop your backpack someone may steal it.

Which wouldn't happen if you were wearing it.

if you wear it during combat, it may get damaged.

It's actually less likely to be damaged if you're wearing it, unless you leave it at home or take the time to hide it out of the way, and in-combat rounds are precious.

it may also slow you down.

Only if it puts you over your weight limit, and the entire point of a haversack is so that is less likely to happen.

under real life assumptions, it's generally better to drop it. the rules specifically handwave the issues of fighting with a backpack. anyway, it depends strongly on how likely it is that your backpack is stolen. if you fight a gang of thieves in an alley, then yes, if you drop them someone may just steal them while you're busy. if you drop the backpack in a cave while fighting owlbears, it's generally a smart move

Except there is no reason to drop it and every reason not to; see above.