Does anyone even use the bitcoin q / core desktop app anymore?
Many people are still using Bitcoin core wallet, including everyone who runs their own bitcoin node and have full blockchain downloaded.
This is the best option if you don't want to trust any SPV wallets and other people nodes with your transactions.
Mobile apps. Which is a good android client for bitcoin?
I would only use android wallets for keeping small amounts of pocket bitcoin money, and many of them are closed source that I don't recommend.
Electrum and Mycelium android wallets are open source, but I prefer how Blue Wallet works.
I have lots of very old (2013) private keys generated on paper wallets from bitaddress.org with bitcoin on them. How do I go about importing them? Is there a good mobile app for importing older non-passworded private keys?
I would not import those private keys in any mobile wallets, better use offline Electrum wallet (you don't have to download complete blockchain for that).