In this situation, our coach said if ball was thrown to 1st, he
thought it would be a force on Adam. I feel like first baseman still
need to tag Adam if i turns back.
The MLB rulebook has this definition for a force play:
A FORCE PLAY is a play in which a runner legally loses his right to
occupy a base by reason of the batter becoming a runner.
The batter received a ball and remained at home plate. Since the batter has not become a runner, there is no force play. Ball is not dead and runners may move to another base at their own risk. Runners need to be tagged out.
But what if Adam gets to 2nd, can the catcher step on home plate for a
force on Ceasar, since Bert should need to run for 3rd?
No. Bert is still entitled to 2nd base (there is no force). So there are no outs to be made on home plate or third (or on first). They need to throw to second. 5.06(a)(2) tells us that the following runner (Adam in your example) would be out if tagged, if both Adam and Bert are on second.
5.06(a)(2) Two runners may not occupy a base, but if, while the ball is alive, two runners are touching a base, the following runner shall
be out when tagged and the preceding runner is entitled to the base,
5.06(b)(2) is for force situations and changes which runner would be out when tagged if two occupy a base.