1. Have tangents in trigonometry got any connection with tangents in circles?

Yes, there is a connection. Click here to see what it is.

2. If you use trigonometry to find one of the angles in a right-angled triangle, how do you find the other?

Easy, just take it away from 90°. (remember there is a right angle, 90° in the triangle).

3. What is one of the most common mistakes made by people using a calculator in trigonometry problems?

They have the calculator set on RAD or GRAD instead of DEG (degrees). 
Use the button marked DRG to ensure the calculator is set on DEG.