Well...
To get a transformer to saturate, you would need to drive it near the edge of its ratings (I would think). This might happen in a tube amp with an undersized OT when it's cranked up. So for example you might get this effect with an underpowered interstage transformer within a tube amp. But think about it... your initial guitar signal is in the range of hundreds of millivolts and very little current (since the pickup is pretty high impedance). I doubt that any transformer would do what you think you want with such a tiny signal dribbling through it.
I doubt it. Assuming you could get a lame enough 1:1 transformer to give out when slammed with a 500mv guitar signal, I don't think you would get bass such as you describe. In my limited experience, when bass F's distort thru a less-than-adequate OT, you get a mushy, squishy sound. Pretty cool if that's what you want, but NOT (to my mind) "rich, full and packed with harmonics".
Winding transformers is a whole science/art unto itself (from what I've read). It would be simpler to buy a teensy isolation transformer, but I don't think that's what you want.
No, the transformer won't work like that. A 1:1 transformer will give you the same Vout as Vin, and (I believe) the same impedance. Now, an OT in a guitar amp takes a 5k-some impedance and makes it into ~8ohm impedance (for example), but I don't see how you could apply this to a tiny guitar signal. Certainly it would need some kind of driver or gain makeup. I imagine it's much simpler to build a small solid-state buffer if you need to lower the impedance of your signal.
I know very little about transformers and inductance, but I imagine if you read up on everything, you'll see why this has never been done. I don't think it will do what you want. If you want to distort your bass signal, build something that seperates your signal into frequency bands and distort the lower band. I could give you some advice for a tube circuit like this but not solid-state...