Erratic booting and at times does not reboot at all

Your picture is not very sharp as such it doesn't offer anything useful on the logic board. The Top left cell looks a bit off but that maybe the lighting and blur.

Instead lets go through a proper diagnostic process. First lets check your battery by installing CoconutBattery post a snapshot of the main window here for us to see Agregar imágenes a una pregunta existente.

Next, lets open the Console app (in Apps > Utilities folder) do you have any current crash reports? if you do safe it to a PDF and post it as well.

Hi Dan,

I am unable to log in. As I log in, it shuts down and any attempts to reboot results in no response. After a time lag and many retries, it will reboot and then immediately shut down. When it boots I see the battery at 36% and once I saw at 95% so there seems to be some battery charge left.

OK, lets try this, disconnect the battery hold the power button for a good 2 minutes, then plug in your MagSafe charger did that stabilize your systems rebooting?

I got it to boot once cleanly and was able to log in and then when I disconnected power it shut down and have not been to reboot it since.

I had to reconnect battery for now to close the lid. This is Mid 2012 MacBook Air 11” [I had wrongly selected as 2011]

I’m also hearing a lot of fan noise, which I could not hear before even though I have not been able to boot it after the one power reset.

Repeat the process and this time with the battery re-installed run this app CoconutBattery and get a snapshot of the apps main window and paste it here for us to see Agregar imágenes a una pregunta existente. You'll want to have a USB drive ready with the app and so you can pull of the screenshot off.