Flyback transformer construction issues
I did debate about this answer being a comment so don't bust a gut voting down (or up) but I had too much to say to fit a comment.

Can you get 3A from those batteries? For now I'll assume you can...

Assuming that you are producing no-more than 80V on the primary (because the FET will break if you are doing a 100v) and given your turns ratio of 1:160, and given the "imperfect coupling factor" (that I pull out of the experienced side of my brain) I'd say you should produce about 5kV.

The FET is getting warm and this isn't a worry but it does rather hint at your 12V supply rail falling to somewhat less when it is powered on. Maybe you can measure it. My suspicion is that it might be falling to about 6V or 8V and this will not efficiently turn the fet on and this could be causing the FET to warm. If this is true then maybe you might only be producing 1kV.

Of course, there is a strong possibility that you damaged your FET with over voltage the very forst time you used it. I note on your earlier threads/questions you were advised to use a 200V device with lower on-resistance.

I'll also point out that a lot of these circuits are from unprofessional sites that might have (with luck) got something to work and easily misconstrued their own circuit diagram thus misleading everyone who tries to build one. Also notable about the website that contains the circuit is the lack of information about the transformer and how to wind it.

The original design came from: -

http://www.geocities.com/CapeCanaveral/Lab/5322/fbt2.htm

But don't bother trying the link because geocities is no more.

Looking at your pictures and knowing how difficult it is to get a step-up transformer working with ratios over 100 I have to point out that there is no hope for the way you have it connected. 555 needs to drive the transistor which provides voltage to the FET and the FET source, emitter of the transistor and 0V of the 555 have to be "rigidly connected". The currents you might be inducing in 0V connections could easily mean you get less than 200V out of this design.