well MY suggestion is when adding a back up camera, BUY the largest screen you can fit, unless you have really GOOD young eye's
as , things get tiny fast on smaller screens
as for where to mount it, that all tends to come out to the lay out of the cab and where its possible and how creative you are with mounting idea's
on a few things over the yrs I helped add them too, we went from using the simple peel and stick double sided tape on some things, to hardware mounted with brackets either supplied or homemade , some were made with hinges to be able to fold down when in use and up out of the way when not needed
Last one I helped install, we used a larger earth magnet glued to the monitor, and then it could be moved to a few spots, and again, out of the way when not in use, as some times, trying to have them in easy eye sight, isn;t easy, and I found if there not in your basic sight view, you DON"T use them as much,as when they are in line with your eye's more!
Just a heads up NOT all monitors can handle having a magnet on them,! , ask the OEM before using one on it if you go that way, found this out when one stopped working we did as such, HAHA!
Now, Back to your Cig lighter, I know on some after market sockets, they actually have fuses on the bottom of the plugs, as in remove socket and at base or inline just below it is a fuse!, so might b worth pulling it to see? maybe some where along the yrs someone swapped the OEM one out?
If you have to replace the outlet(or just plan to install another one)
I really like the one's that light up, makes finding them easier if you work at night in them, or sure does for me and my old eye's!
plus the new one's now have the phone charging ports , I recommend one's with covers, typically marketed as WATERPROOF one's!, this way you get or MAYBE get better hardware that won't rust out as fast or just having a cover can keep dust and dirt out, as well! any Good marine parts place would carry decent one's , or sure they can be had on the Ebay/amazon like sites too