The statement 'there is no speed limit on the Autobahn' is not really correct. I couldn't find any official statistics, but I've seen a number of only 50% of German Autobahns without any limit.
Many parts have a permanent limits of 120 or 130 km/h (74 mph or 81 mph). Some parts are limited to 80 (50 mph) due to construction work.
Also, some limits are only for wet roads. Finally many parts have dynamic speed limit signs that change based on the traffic situation.
Even though there is no general limit for the 50%, there is a recommended speed of 130 km/h (81 mph). If you are going much faster and there is an accident, it is more likely you are blamed for it (if you survive it, which is unlikely).
Autobahn police can still pull you over for reckless driving or driving too fast in certain situations even when there was no limit.
This really depends on the situation, there are some stretches in less populated areas that get relatively light traffic and you can drive 200 km/h (125 mph) with no problem. It also depends on the time of day. There is rush hour traffic that limits your speed anyways. Late at night it is usually empty and you can go fast.
It also depends on you car, it makes a big difference if you go 200 km/h in a big BMW or Porsche or a smaller older car. It is also much more tiring to go fast, you have to concentrate much more at 200 than 120 because any tiny mistake has much bigger consequences.
I had several friends visiting from overseas, renting a car and then were too scared to drive really fast. If you are planning to go fast, start with a speed you are comfortable with from home and then increase it over time. Don't go 230 km/h (143 mph) as soon as you hit the Autobahn for the first time.