Most of your questions are answered at http://homedistiller.org/methanol.htm
Yes, along with ethanol (good), the yeast can make small amounts of other organic compounds, such as methanol, propanol, butanol, acetone, ethers .... etc. Literally thousands of them. Thankfully, most are in the smallest proportions, only detectable with advanced testing techniques.
So all these are already present in your beer, wine etc.
The trick to make less of them is to keep your yeast happy while its fermenting - eg keep the temperatures in the right range (below 25C), not too high in sugar or alcohol concentration, right nutrient levels etc.
Distilling gives you the opportunity to separate them out, so that you can get clean ethanol, away from the acetone, methanol etc. That is if you're after a clean vodka. Its these same compounds that contribute to the "Taste" that we call rum, bourbon, whisky etc, if present in the right proportions.
As you begin distilling, the temperature in the boiler and column will increase as you add more heat. You will find that the various compounds tend to come off in the greatest proportions around their boiling point. So as the pot gets into the high 60's degrees C, you'll find the "foreshots" to contain most of the methanol and acetone, and hence can easily be separated and thrown away (as a rule of thumb, toss the first 100 mL you collect per 25L of wash boiling). The clean ethanol tends to follow, around 78C. After a while, as the amount of ethanol present decreases, the temperature will rise more, into the 80-90, and you're into the flavoursome "tails". At this point you stop distilling when you find the flavour is too much for what you want.
The real risk of poisoning comes from your good ethanol - simply having too much too fast - stock standard piss-as-a-newt alcohol poisoning as experienced by most high school students.
The risk from the methanol etc is highly over-rated. First, to consume it, means that you haven't followed about the only rule written in stone for this hobby - throw away that first portion you collect. To drink it would be bad enough, as it really stinks - you know its wrong. For the trace amounts that reside in the final alcohol, the proportions are far far too low to worry about.
Likewise the tails. These mostly taste bad, rather than be poisonous. For the levels present, you cant poison yourself. The link (above) goes through the numbers on this, if you want to get into the facts in a bit more detail.