Well, that's strange!
If your voltage is showing mid-14's then the alternator and regulator are working properly.
With the lights in the dash going out I'm going to guess that you don't have voltage to the ignition switch and, therefore, it can't pull the starter relay in. Why is why it won't start.
But if the lights come back on by themselves it seems like something is getting hot and losing connectivity. Then cooling and connecting again. I don't remember having a bad connection like that, but that is my guess - a bad connection at the battery or starter relay. Check your connections there, including the grounds.
Gary, AKA "Gary fellow":
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Dad's: '81 F150 Ranger XLT 4x4: Down for restomod: Full-roller "stroked 351M" w/Trick Flow heads & intake, EEC-V SEFI/E4OD/3.50 gears w/Kevlar clutches
Blue: 2015 F150 Platinum 4x4 SuperCrew wearing Blue Jeans & sporting a 3.5L EB & Max Tow
Big Blue: 1985 F250HD 4x4: 460/ZF5/3.55's, D60 w/Ox locker & 10.25 Sterling/Trutrac, Blue Top & Borgeson, & EEC-V MAF/SEFI