Assuming the person turns themselves in, but is not acquitted of the
charges against them after talking with the police, they then must go
through the process of being charged.
For what it is worth, police might not pursue charges or might dismiss the charges, but an acquittal is a legal term of art that happens only after a judge or jury makes a ruling on the merits in a case, based upon a trial or a motion from defense counsel (generally speaking, during trial).
An acquittal provides double jeopardy protection against a retrial and isn't subject to a direct appeal (or more accurately, to a direct appeal that impacts the outcome of that particular case; a few states allow direct appeals from acquittals that have no impact on the defendant, but impact the rule of law allegedly misapplied in the case in future cases).
While in the trial for these charges could this person bring a
counterclaim against the police for lost wages
No. There are no counterclaims in criminal cases.
or could this person (after being adjudicated) bring a claim against
the police department in civil court?
To prevail in a civil lawsuit against the police, the person would have to show that the claim was brought without probable cause with the intent to deprive that person of his constitutional civil rights (i.e. that the police sought and obtained the arrest warrant knowing that there was no probable cause to bring the charges or issue the warrant).
Mere dismissal of the charges, or an acquittal at trial, does not generally entitle a criminal defendant to any legal relief (a few states are exceptions to this rule, but not many).
A civil lawsuit against the complaining witness would also be subject to at least a qualified immunity and would generally require, at least, proof that the complaining witness intentionally made a false report to the police when it was foreseeable that this would harm you. The liability of a complaining witness would be mostly a matter of state law rather than federal law.