Yes they do, but cash deposits for basic accounts can be expensive for some while it’s free for others. You would need to compare the prices in detail. Also they will often refuse to open a basic account for persons that already have another account (incl with different banks). And most banks (all i know) refuse non-residents for checking accounts.

In case you know a bit of German you can search for Bareinzahlung auf Konten Dritter.

Postbank does not offer the service any longer. Apparently by now only the Reisebank does, charging 1% of the amount transferred, ar least 7,50€.

They all will ask for proof of origin for the money, eg bank statements of the withdral, when the transfer is more than 2500€.

Since they cooperate with western union, sending money from/to EU members should not be an issue, but I could not find a specific answer.

German post offices would typically not provide financial services beyond putting cash in an envelope and sending it to a mailbox. The financial services of the Deutsche Post was split into Postbank which by now is a part of Deutsche Bank. Asking for cash deposits in a German post office, staff will likely send you to a bank.