Can a landlord charge tenant eviction fees and demand it paid before stopping the eviction process?

As you wrote in your question, it appears that the landlord has already paid the J. P. Court the filing and process of service fees. These are considered recoverable damages, depending on your contract. In most instances under Texas property code §92.019, late payment of rent, fees, the landlord must abide by this rule. Since the landlord is claiming that you broke your lease by not paying your rent, normally Justice of the Peace Judges will add the court costs to the amount owed to the landlord. Your Landlord does not have to accept the rent and late payment only. You are being given an OPPORTUNITY, deal, not to have an eviction on your credit, and having the landlord evict you. The additional fees are minimal compared to coming home one day and finding your possessions removed. The Landlord has paid these fees and is now entitled to the rent, late fees, and court costs. My suggestion is to immediately pay the landlord.