To see your exchange history, open the left side menu, select any of the exchanges listed, and go to "... / Exchange history"
If an entry in the history is still "waiting for deposit", it's possible the status wasn't refreshed automatically. You can open it individually and select the "refresh" icon at the top right corner. You can also check the status manually on your browser following the instructions here.
Coinomi does not store any user information on its servers. There are no central logs or backups of your exchanges. Instead, the exchange history is stored locally. If you reinstall the application and restore your wallet using your Recovery Passphrase, you will clear your exchange history. The same is true if you restore your Coinomi using your phrase in another device (simultaneously or not).
Note that your transaction history for each restored wallet is not deleted. All your outgoing transactions that were "exchanges" can still be seen there, even after the Exchange history is cleared.
We will add an option to export your Exchange history in one of our future releases. We're also hard at work ironing out any slight UI bugs of the in-app exchange, specifically concerning the exchange history.