Connections between 2 separate isolation transformers
What you are describing is normal connection for these transformers in cases where they must share a ground path.  The ground connection provides the reference to which the output voltages of the transformers are "attached".  

When you connected the transformers as you've described, nothing will "happen" but you will have two different types of 240V available to you.  You will have the output of the center tap transformer, which is two opposing 120V AC signals which provide 240V line to line, and the output of the non center tap, which is a full 240V line to ground.  This draws to attention the fact that even if two systems are the same voltage, getting 240V from a center tap +/- 120 transformer is significantly safer than from a non center tap.

It is safer because it halves the voltage available to ground faults. You can still shock yourself with 240v line to line, but a line to ground shock will only be 120, which is still quite dangerous (little burnout damage but still significant heart risk). The same applies for non-human-shock faults. If you had to have a flash from a short to ground of your choice of 120V, 240V, or 347V 1 foot from your face, which would you choose? I've seen all 3 in person (347V will give you a mild suntan) and given the option I'd go for the 120V flash or none at all, every time.

If you measured the outputs of the transformers to each other, you would measure 0V ground to ground and if you measured Voltage between all of the line outputs, you would find the non center tap transformer was 120v to one of the phases and 360V to the other phase of the center tap transformer. 

As far as your additional question, the power company will be unwilling to hook up an unsafe transformer for you, but as far as the downstream connections it will be you or your electrician and your electrical inspector that ensures the safety of that.  When connecting a transformer to the power network, you must ensure that the input ratings are correct and that the turns ratio is correct for the output voltage you want.  The output can be any acceptable voltage (Center tap or not).  If you wanted to parallel the outputs of the two types of 240V transformer, you must not.  Transformers used in parallel must be carefully matched, ideally identical. 

When you ground one of the outputs of an otherwise isolated transformer, you are simply selecting which will be the identified conductor(neutral).  If one output is grounded it is important that the other output never be connected to ground as in most cases this would be a powerful dead short.