I generate the password and stored them in my password manager under the notes. 1Password added functionality seemingly recently to add security questions and generate a random word string that I use these days.
Note that you should not generate a random password like D27fX$0f7RyD for your security questions. These are designed to give to a human operator on the other end of a phone. If an attacker calls up the account recovery line, gets asked for a security question, and just says "heh, I think it was a string of random characters", there's a decent chance the human operator will let them into the account. As you say, use an actual word string (passphrase) generator, which is a bit less susceptible to this attack.
Yep, if you can choose the question, choose something like "What was your first pet's name?" and then make up something silly like "Mister Poopy Eyes" (a conceivable child-given pet name).
My work provides me with a 1Password subscription (for both work personal use) that I take advantage of that is pretty good. I think they only require you to reauthenticate with your master password once every two weeks or something. I use a PIN, biometrics, or my Apple Watch to unlock it when it timeouts in between that two week period, and I've had no problems syncing between several of my devices.