You can do the agarose gel without EtBr and use this agent in a solution only after the electrophoresis to visualize bands under UV light. You avoid to contaminate all buffer and electrophoresis vessels with that toxic agent.
It is added because you can visualize the DNA bands at any time, and if the sample needs to run longer you can just put it back in the electrophoresis tank (no post staining required). However, it will change the migration of scDNA, so for that I will stain afterwards. Midorigreen (Bulldog Bio) works great and is not toxic.
Most agarose gels are so thick it takes too long to get the EtBr to diffuse into them. The same problem exist for other stains with agarose. Adding to the gel saves time. I add a little more EtBr to the volume at the positive side for the chamber because EtBr moves toward the negativity during the run, which depletes it on the the end of the gel furthest from the loading wells.
As mentioned by Francisco it has to be added at the end of the run to avoid contamination. Many other dyes are available in the market to replace etbr, like Safe blue stain, syber green.....
To add EtBr as you prepare the gel is, in my opinion, faster than staining the gel after it has ran. EtBr is not supposed to affect the migration of DNA dramatically, there are some old articles that you can review on this regard. I also use acridine orange as a less toxic option, but it produces too much background. The option that works the best for me is SYBR Safe (Life Technologies), you should be able to find other dyes with similar advantages to avoid using EtBr.
Etbr is usually a fluorescent tag to tag the DNA molecules according to the base pairs. intensifying almost 20-fold after binding to DNA. etbr could be replaced with SYBR bt the fact is all the alternative dyes are more mutagenic than etbr and hence we prefer etbr.
When we used Etbr, we stained the gel after the electrophoretic separation to avoid contamination of the electrophoresis equipment as Franscisco pointed out. To avoid any toxicity and the problems with contamination and disposal of toxic substance, the entire department changed to Midori green (sold by several companies). It's less toxic and you can use the same equipment and filter settings for detected as for Etbr.
There are commercial patches containing Etbr, which can be applied on the surface of the gel after electrophoresis for 5 mins, then use UV to visualize the bands. This is the safest way to use Etbr.
The SYBR stains often cause a lot of band-separation compression because they bind tightly and are attracted to the opposite pole as the DNA. It might not be a problem for most, but for me where I attempt to separate based on a 1% difference in length, they are unusable. If you find compression a problem, you might consider this property.
You can use EtBr for the visualization of DNA bands by staining of a agarose gel(made without EtBr), even you can use the EtBr directly to the samples before loading in to the wells, but all these methods are time taking ,inconvenient and do not give better visualization compared to use EtBr while preparing a gel.
Etbr is used in concentrations of 0.1-0.5ug/ml in DNA-gels. In this concentrations Etbr is not mutagenic in Ames test. Since you are not eating the gel nor drinking the buffer the amount you possible incorperate is very low. Its probably more toxic to smoke a cigarette then taking a bath in a DNA-gel buffer. However, you should open the stocksolution under a hood to be save.
Ethidium Bromide is an intercalating or stacking dye that enters the purine and pyrimidine base pairs. Therefore, it is carcinogenic. It is preferable to use ecofriendly flourescent dyes such as SYBR dyes.