Ok, so a dragon ate Sayma this morning, will her shops get a new owner or is Bits and Pieces closed for good?

According to the uesp, it will be closed for good.

go to another hold, eait for a week or 2 then go back to bits and peices and check if anyone has took over.
if no one has then you have your answer.

Source ?
Looked around UESP , didn't find anything about closing the shop.

No one else is mentioned. I assume she's all there is.

Sayma is the only owner of Bits and Pieces, so once she is dead, the shop is gone for good(as far as I know this case is unique to Bits and Pieces, there isn't such a thing for any other shops in the whole Skyrim).

What I'm really wondering is, how could Sayma be killed in a dragon encounter? Does she even go beyond city walls?

Dragons (very, very, rarely) attack inside the city.

So if she's dead, is it possable to use Bits and Pieces as a base of operations in Solitude? Not saying to store stuff there or sleep. If you take (ie steal) the items, will they respawn later?

To be honest I'm not sure if the dragon did it or if I did when I swung at the dragon. I'm pretty sure it was the dragon because I didn't get a bounty.

As for bits, damn. That's dumb, we need a fix for this beth. Maybe patch it and let Gisli have it.

It actually happened to me twice, during two different gameplays. But it happened the same way and it was the same place on both of them, in Riften. Once I entered Riften through Honeyside, a dragon appeared and started to attack the city. But I guessed it might be because Honeyside is not a part of Riften(it shares its background theme with outside world, not with city background theme) a dragon could have followed me to the city from outside of the Rfiten walls.