The term "probie" is short for "probationary", the period of time that Brian and Tim will spend learning the techniques and safety rules and other factors about the job once they've graduated from the academy. They may have finished their academy work but they still have to learn a lot and prove they can handle the rigors of the job. They are also referred to as "recruits" and "candidates". The same sort of system is found in other professions of a dangerous nature like doctors, nurses or police officers though the terminology differs in each.

Steven probably feels that the mask is too much of a hindrance to his job performance. The mask does supply fresh air to the user but it also can limit the range of their vision and might feel very stuffy and uncomfortable. So Steven refuses to wear his.

We also see that Steven is driven by his ego and his need to be a hero. He tells Brian a few times that he won't die by being burned or otherwise killed in a blaze and that their father was his inspiration: their father was also a well-known and highly regarded hero of the Chicago fire department but didn't have the ego issue that Steven does. The bravery necessary for the job is something Steven feels very strongly about, however, it often puts him directly in the path of danger along with his comrades.

Steven took that attitude with his brother because first, they're brothers and siblings are often harder on each other. Also, Brian had washed out of the firefighting academy a few years before, which had to be very disappointing to his older brother, who must've had high hopes and obviously thought Brian would join the family tradition. Steven was likely also, despite his amusement, angry with Brian for trying to bribe the city fire commissioner to assign him to a firehouse far away from Steven's house, Engine 17. That definitely added to the friction that has dominated their relationship for many years. So Steven, thinking his brother might wash out as a candidate, made drilling tougher for Brian in the hopes that it might toughen Brian up to the job or prompt him to leave.