Is the contract concluded between the father and the lawyer considered valid, and is it permissible for the lawyer to demand the agreed-upon amounts from the questioner after the case concludes without a time limit, especially given the dispute over the details of the original agreement and the perceived amounts?

As long as you have appointed your father as your agent to conclude a contract with the lawyer, and you have affirmed your acceptance of whatever your father commits you to, then you are bound by the material obligations and expenses that your father undertook on your behalf. Shuraih al-Qadi said: "Whoever imposes a condition upon himself willingly and without coercion is bound by it." You are permitted to withdraw from the ju'alah (reward-based contract) before the lawyer completes the case, but you must pay him the customary wage for the work he has done. It is stated in Matalib Uli al-Nuha: "And after a worker has begun the work, if the one offering the ju'alah rescinds, then he owes the worker the customary wage for his work."