I have a mid 2017 MacBook Air, running the latest version of Mojave, OS 10.14.3. I don't use Time Machine for my backups (I prefer SuperDuper!), but regarding your statement "the Time Machine logo on the menu bar doesn't circle when updating", do you mean when Time Machine is doing a backup, or when a newer version of Time Machine is released (and thus expecting an update to be installed)? For the latter, don't think Apple releases separate updates for Time Machine (unless it has a serious bug). Such updates are typically included in a newer version of the Mac OS. Not sure about the former.
Regarding the "need" for entering a password every "so often", here are the results of a google search of "How to turn off the entering of a password on a Mac":
I actually require the entry of my admin password on each of my Macs when starting up. But neither machine makes me re-enter it after it has been "sleeping", or I have been away from it. In fact, when I do my weekly SuperDuper! backups, I quite often leave the backup alone while I am doing other things (primarily continuing to clean our place), and when I come back and "wake" the machine up, it continues with the SuperDuper! backup, and I am not asked to re-enter my admin password.