I understand your fascination with "combustion chemistry" - after all, that's what originally got me interested in chemistry many years ago. Unfortunately, the chemistry involved is typically messy and poorly understood, which may not make the "explosive polymerization of 4-nitroaniline reaction" a very good final project. Certainly the materials involved are quite hazardous: concentrated sulfuric acid (as you surely know) is a highly corrosive oxidizer even at room temperature, and 4-nitroaniline is toxic, harmful to the environment, and a suspected human carcinogen

A hood alone is not enough protection though, you will also need the appropriate personal protective equipment, meaning safety glasses, a lab coat and gloves. Disposal of the generated waste may prove to be the biggest challenge, because it is unclear what exactly the "polymer" is that the reaction generates. The elemental composition that is given in the article you referred to tells you that it is not, as the video states, "pure carbon". So the polymer has to be treated as if it were as hazardous as the starting materials (combined now - so you have toxic, corrosive, environmental hazard, and potential carcinogen) and you need to dispose of it accordingly. This means handing it over to someone/a facility who can handle such waste. If your high school does not have a procedure for handling such waste, you could probably inquire with a college or university near you and they might help you out.

There is no way I would consider it safe or acceptable to dispose of the waste from this experiment in the regular trash. As a side note, will the final project involve some sort of scientific inquiry - investigating effect of the ratio of reagents on the outcome (amount of foam maybe?) or whether different nitroaniline isomers show the same/similar reaction?