I don’t really know what your agency would be expecting but I have looked at a lot of audition tape and selected actors by it. I know what I have generally seen from audition tape and what worked for me when I was watching tape.
I think dressing up too much like the character would be overkill (for instance, wearing the toga) but you might want to do some creative thinking in what they might want the character to look like since in a video audition a visual impression is important. I think whoever is making the decision will want to see that you have the physical potential to fit the character- this is an easy thing to tell whereas sometimes judging performance on tape is harder. For instance, for this “nerd” character, you might wear glasses if you have them, or make sure your hair is in a ponytail, or something simple like that. Don’t go overboard and look ridiculous- but it can work if you’re subtle. I would try to think like the character and where what they might wear out of your closet. Make sure you feel comfortable and your performance will be better.
For the video itself- you should probably just have someone read with you and feed you lines from off camera. Sometimes I’ve even seen agents read with their actors from off screen. It’s you’re audition, so whoever you’re reading with should not be on camera. Make sure their lines are audible too however. Sometimes if the offscreen character acts too much it can be distracting but i think it’s better if they’re not totally flat so you have something more to react to.
It helps to have someone work the camera (not too much!) and give a little shot variety. When I was having recorded auditions sent to me I always asked to make sure that the camera person had both a medium shot and a close up. The close up is key if it is a film role especially- a medium will be good for showing the actors body language.
Most tapes I’ve seen were just with the actor seated comfortably in a chair- but if you want to move about and have a full shot it might be ok- just don’t have hand held camera work following you or anything- keep it on a tripod I think and keep the camera work very simple.
Those are my thoughts- just from my experience- there may be different expectations for what you’re doing- so I don’t want to lead you in the wrong direction- but maybe this can help.