3.) 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?
4.) Maybe I should use a desktop client to import the keys in a safe linux machine rather than android? Is there an "online" version of a desktop app so I don't have to download the whole blockchain?
NeuroticFish is right here - you should only import your private keys on to a clean OS with a freshly installed wallet on an airgapped computer, to maintain their security. The best way of doing this is to use an old computer or laptop, disconnect the WiFi card and unplug any ethernet cables, format it, install a Linux distro of your choice, then download and verify Electrum on another device and transfer it to your airgapped device using a formatted USB drive. Alternatively, after airgapping you can boot to Tails from a live USB which has Electrum pre-installed. You'll need to create a complementary watch-only wallet on an internet connected device which contains only your addresses so you can view your balances and create unsigned transaction, which you then move to your airgapped device via a USB drive to be signed by your private keys.
However, the main question is "What do you want to do with any coin that is on these private keys?" If you just want to see what you have but not actually spend it, then I would suggest not importing the private keys anywhere and keeping them on the paper wallet. The paper wallets should also have an address you can look up to view your balance.