1.) Does anyone even use the bitcoin q / core desktop app anymore? I remember I'd always have the whole blockchain downloaded and run the official client, but it seems that people favor "online clients" these days if that's correct terminology? Blockchain online but wallet stored on client to save room? I figure that's a must for mobile apps now?
Although maybe not necessarily on mobile, SPV wallets have a big "market share".
But the later news made more people realize that they can be tracked and some more may have started using Bitcoin Core again (or their own servers, which need Bitcoin Core).
Also, let's not forget that far too many users (still) keep their funds in custodian wallets, usually exchanges.
2.) Mobile apps. Which is a good android client for bitcoin?
I use (rarely) Mycelium. It has (plenty of) room to improve (to be overly nice), but it's not a bad wallet and it's one of the very few that support hardware wallet, which imho it's a must, especially on mobile phones.
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?
Mycelium can do that. But.. just read forward
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?
Yep. Safest would be to not import them into anything that's online. If it's big funds please stay on the safe side.
Learn the basis of cold storage and import the private keys onto a system that will never go online (it can be even a Linux on an USB stick) and keep only the addresses in a view wallet online. It's enough for spending safely.
Also consider buying a hardware wallet.
5.) Is it any more or less secure using online clients vs having the blockchain fully downloaded?
If you use SPV clients your funds can be tracked to you.
The coins can be stolen from you with or without Bitcoin Core if your system is not secure or you make mistakes (like keeping the keys "safely" in the e-mail)
I mostly tuned out of all latest updates, releases, security vulnerabilities since around 2015. Is there any news I should be up to speed on? I'm still trying to wrap my head around DeFi, NFTs, smart contracts, and all sort of new protocol lingo. Also Lighting/L1?
I'd start with cold storage, hardware wallets, SegWit and Lightning, especially as you have bitcoin.
Then you can move on to altcoins/DeFi and such, although keep in mind that many of those may be just money traps (you can read about ICO era).