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?
Some Bitcoin enthusiast including me still use Bitcoin Core.
2.) Mobile apps. Which is a good android client for bitcoin?
Blue Wallet, Electrum and Mycelium are good option since they are non-custodial and open source. But if need wallet with good UI/UX, i would recommend Blue Wallet.
5.) Is it any more or less secure using online clients vs having the blockchain fully downloaded?
For wallet which require you to access the website, there's risk you're accessing fake website or installing browser extension which could steal you data.