When you titrate an acid with a base or vice versa, you change the pH of the solution in the conical flask. The indicator in the conical flask will change colour according to the pH and this will allow you to determine the endpoint of the titration.
To determine the excess reacatnat for the second part of the back titration, you add a standard solution of a strong acid/base to the solution from the first part of the prac until it reaches the end point. Then you do the n=cV stuff and so on.
You can fill a burette with an acid or a base, as long as the solution in the conical flask is the opposite pH.
Diluting a solution would reduce the difficulty of the titration and increase the accuracy. If you titrated a super ultra mega concentrated acid/base, you might reach or pass the end point in one drop.
Safety is overrated.
Errors in the equipment can occur when you rinse it out with the wrong substance. You must rinse out the burette and pipette with the same substance as you are going to use in it. The conical flask should be rinsed with distilled water. I think rinsing the equipment with the wrong substance would introduce a systematic error.