This is becoming confused, slow down.
The pool that holds the funds that you sent from the other wallet is designated primarily by the address that the wallet sent to i.e. if you sent to sapling, for instance, then thats where they are.
This mixing you talk about is either done by the pool transfer tool or multisend function in Ywallet. Otherwise, most wallets only support sending to a single receiver address in a normal tx. But yes, transparent, sapling and orchard accounts are all under the single seed. Private keys are the legacy (sort of) method and only pertain to their specific address/account and pool.
Zashi is still beta and probably not the best example, but ideally yes, it is very likely. However, given inherit differences in wallet functionality, seed restore error is something to be mindful of. Without prior assertion that your backup will absolutely work on some other wallet platform, dont assume it absolutely will.
A note in your backup containing the name of the wallet from which it was derived (as well as a birthday height!) can you very useful. Especially if, for example, the wallet is no longer available or restoring to it is somehow not an option.
Wallets based on the current version of the ECC SDK do not support Orchard. Therefore if you restore a seed in one of these wallets, it would appear that coins are missing.
They are not. Once the SDK supports Orchard and the wallets are upgraded, your coins will become available. In the meantime, you can use Ywallet or Zingo.
When you send to a UA that has both Sapling and Orchard receivers, Ywallet splits the amount equally to let you send to either pool with full privacy. If you want to target a specific pool, you need to use an address that has only one receiver or a native address for that pool.