I'm on the the verge of buying a Moto G from Motorola for my girlfriend, but I would like to know if it is possible to run Flash on it. I can remember that I had it on my Nexus One a couple of years ago, but currently I can't seem to find it on my Galaxy Note 3?
Android (Google) dropped ALL support for Flash Player last summer. Adobe continued to update security fixes for older versions, but that ended Spetmeber 2013. There is, and will be, no more Flash Player for Android devices going forward.
Android recommends "Dolphin" or "Puffin" (free in the GooglePlay store) as browsers to view Flash content. They use "server side" rendering to reduce the load on the device and preserve battery life.