As the old saying goes, if you are cold go find the cat. She’s already found the warmest spot in the house. My childhood cats used to fight over a hammock we had hanging over one of the radiators, it never ceased to amaze me how long they’d spend in that thing and the heat never bothered them. If you are concerned kitty is going to burn herself, invest in a fire guard which is a mesh barrier usually used to stop toddlers getting to the fire but is also effective for curious cats.
Blinking check engine = pull over and call a tow truck. You’ll do even more damage by continuing. As for shaking, it could be a bunch of things and I’m not going to speculate beyond what other people have commented.