In a Civil Trial, how do courts handle scenarios when evidences presented are denied (by plaintiff/defendent) as forged, tampered, or claims that 'I did not sign it' or 'That is not me'?
Just like every other contested assertion
They hear the testimony (evidence-in-chief, cross-examination, redirect), look at whatever physical evidence and submissions on the law that the party wishes to make. The other party will do the same. They weigh the evidence including such vital considerations as to whether they think one party or the other or both are full of shit and, if the evidence is such that the party has proved their assertion on the balance of probabilities, it is now a fact which bears on the judgement.
It is not unusual for there to be expert witnesses (potentially on both sides with opposite conclusions) on the subject of handwriting analysis testifying on matters like this. They will go over known samples of the person's handwriting, and compare small bits of the writing to what is present in the document in question.
Keep in mind that prior to trial, documents will be the subject of written interrogatories, requests to admit, depositions, summary judgment and motions in limine. If it is alleged that a document which purportedly was signed by me in Denver on January 1, and it is undisputed that I was in Shanghai on that date, a motion in limine to prevent the document from being used or reference at trial would lie.