How to Franklin not lose the game because of Lamar

For the Bail Bond missions in particular, and probably other missions you may want to track, you can view your progress by signing up with Rockstar Games Social Club and linking the account you're playing with.

In the Social Club page for GTA V, you can view your Career, and a tab under that category labeled Checklist will allow you to see all your progress in various missions. Maude's missions are under the Hobbies and Pastimes section.

This will only tell you whether you have completed the mission or not, though. It does not indicate where the target is located, and unfortunately this information does not appear on the Random Events and Strangers map at the bottom of the Checklist page. I suspect that the map could still be helpful for you to find other sidequests and events you haven't played yet.

As far as I know, there is no way to get back the emails or texts with mission information once they have been deleted (I'm kind of surprised they can be deleted; disappearing spontaneously would definitely be a bug). Resorting to a guide may be necessary on that front.

You can't quick save while on a mission, sadly. You'll have to wait for the mission to end (you get a little summary screen and are generally asked to press A/X to continue) before you can use the phone to quick save.

I've also been in a state before where I couldn't use the quick save feature even though I was pretty sure I'd finished any active missions. In that case, sometimes killing myself reset things. Other times that didn't work and I had to load my last autosave instead.