Do criminal prosecutors, or judges and juries, view charges or sentencing for fighting and/or injuring someone in public differently if the person facing the charges is a trained fighter like a professional boxer?
AFAIK "fighting and/or injuring someone in public" is not a crime as such anywhere in the world. When a person is charged with and prosecuted for a crime, the crime itself is very specifically defined.
For example, a person who fights and injures another person in NSW, Australia, could be charged under the Crimes Act 1900, with
- Intent to cause grievous bodily harm under s33(1)
- Reckless grievous bodily harm under s35(2)
- Reckless wounding under s35(4)
- Choking, suffocation and strangulation under s37
- Assault occasioning actual bodily harm under s59
- Common assault under s61
- Affray under s93C
The extent, if any, that a person's status as e.g. a professional boxer will impact on the prosecution's burden and/or judge's sentence will depend on the particular crime. For example, it probably doesn't matter for a crime of "intent", however, where "reckless" is an element, the fact that the person knows more about causing physical harm to another human being may make behaviour reckless that would not be reckless if performed by someone without that knowledge.
As a further example, the crime of affray includes the phrase "whose conduct is such as would cause a person of reasonable firmness present at the scene to fear for his or her personal safety". A 6'2" heavily muscled boxer is in a better position than a 4'3" 14 year old girl to "cause a person of reasonable firmness present at the scene to fear for his or her personal safety".