Atonement of sins of mankind?

Christians believe that Jesus gave his life for the atonement of the sins of mankind.
a) Does this mean that by his death, he atoned the sins of all people born before his time only or even of people who were born after his time?
b) Does the death of Jesus on the cross mean that the sins of all Christians would be forgiven, even if they continue to sin and disobey God’s commandments?
c) Is there any accountability for Christians for their personal actions (good or bad) on the Day of Judgment, irrespective of the death of Jesus on the cross?

I will answer your questions in the order you asked.

1. In Hebrews, the sacrifice of Christ for the sins of man is referred to as the “once for all” sacrifice (Heb. 7:27; 9:12; 10:10). This means that all atonement prior to Christ was made possible by looking forward to that “once for all” sacrifice that Christ would make and all atonement since Christ is made possible by looking back to that “once for all” sacrifice (cf. Heb. 9:11-15). As the “once for all” sacrifice for sins, He is the propitiation (the atoning sacrifice) for our sins (Rom. 3:21-26; 1 Jn. 2:2).
2. The death of Christ on the cross provided the sacrifice necessary to atone for our sins. However, a person has to meet the conditions of that atoning sacrifice. Jesus died on the cross for the sins of the whole world (1 Jn. 2:2), but a person has to obey the gospel in order to receive the benefits of that sacrifice (Rom. 1:16; 2 Thess. 1:8; Mark 16:16; etc.). If you are drowning and I provide you a life raft to get you out of the water, that life raft doesn’t do you any good if you don’t climb on it. The sacrifice of Christ is the life raft, it is up to us to climb on it through hearing His word (Rom. 10:17), believing what it teaches about Him and His kingdom (Acts 8:12), repenting of our sins (Acts 17:30, 31), confessing that we believe Jesus is the Son of God (Rom. 10:9, 10), and being baptized into Christ for the remission of our sins and to be added to His body, the church of Christ (Acts 2:38, 41, 47; 1 Cor. 12:13; Gal. 3:26, 27; et. al.). Once we have obeyed the gospel of Christ for salvation, we must continue to live according to the gospel to stay in a saved condition (Col. 1:23; Gal. 2:20; 2 Pet. 2:20-22).The blood of Christ continues to cleanse us from every sin IF, under the condition that, we walk in the light as He is in the light (1 Jn. 1:7).
3. Everyone must appear before the judgment seat of Christ on the day of judgment (2 Cor. 5:10), including Christians. The benefit of the Christian is that they will be judged based on their new life in Christ, having been born again (Jn. 3:3; 1 Pet. 1:23). However, having been born again, they will still be judged by the things written in the book (The New Testament) (Rev. 20:12). Yes, there will be many Christian who will be lost because they did not remain faithful to Christ (Matt. 13:47-50; Heb. 3:12-14).

I hope this answers your questions and helps you in your study of God’s word.