A young adult entered a home where he used to rent a room in order to
retrieve a piece of workout equipment... This young man was charged with
theft and burglary and even though he has evidence that the equipment
belonged to him...
This is your assessment of the situation, not the prosecutors' or courts'. Local or state laws may see this entry and property removal as trespassing and/or theft, even when there is a dispute over past rent or the past lease.
I might also note that he was forced to move out without being
That's a completely different situation and is a civil offense, not a criminal offense, and it does not typically give an evictee a right to trespass and remove property from the past residence.
How in the world can this 13 year old be charged under these
The prosecutor with jurisdiction (city, county or state) makes the decision to prosecute on what they see as the strength of the evidence, the severity of the crime and the likelihood of conviction by a jury or judge. Since the subject of prosecution is a minor, other local laws may come into play regarding who in the judicial system has input into the decision to prosecute.
If you feel like the prosecutor is overstepping, tell the 13 year old's parent or guardian to talk to their lawyer (court appointed, or privately hired) and consult with the prosecutor to drop or amend the charges.
Can the prosecution possibly have a case?
The prosecutor must feel they have a case; if the case goes to court, it's up to the judge and jury to weigh the evidence and convict if that evidence is beyond a reasonable doubt; or if the judge or jury feels that the evidence is not convincing, they acquit. Since the subject of prosecution is a minor, other local laws may come into play regarding if the court procedure must be a jury trial, a bench trial, or some form of mediation and/or restitution.