For soldering the only thing you have to do is get it open. All the soldering points are easy to get to without removing anything besides cover sheets. You won't even have to disconnect anything!
As for caps, I don't think they are the issue, but yes one should not rule then out yet. Only because the booting is too random. Usually a component failure causes a more consistent issue. Also it seemed like once it warmed up a bit, it ran. Caps get funky once they warm up, not when cold so much. However cracked solder joints do work better when warmed up from thermal expansion.
I really would not have that thing plugged in until you get a closer look.