Has it occurred to you that the problem might be you?
Now I realise this is an incendiary statement, but you having had numerous relationships, how come it's the same result every time? Maybe you get bored easily, set you standards too high or just don't want to put much work into things?
It took me a number of years before I realised that some crazy behaviour I saw from my partner at the time was actually a reflection back of my own unreasonable behaviour. Sure, they were unreasonable, but that was because I'd been an ar**hole in the first place. I changed my behaviour and now am in my longest ever relationship (by some margin).
Don't get me wrong, some folk are just plain crazy....but not all of them. The examples you gave of your current girlfriend do sound odd, but is there some signal from you that has made her feel so insecure? Do you play it too chill, not putting enough effort in and then acting surprised when she's becomes worried?
You speak about having a dog, but dogs are pack animals and rely on mutual reassurances. Ignore a dog and it will start acting up, crying, following you around everywhere etc. Maybe a 'oh so chill' cat would be better?
Anyway, good luck and hope you work things out