Our company wants to change not using daily time sheets.

It is the responsibility of the employer to keep track of employee's time, and the employer can utilize various ways to do so. If this is how they intend to do it, then you need to comply if you want to remain employed.

Yes, it is legal so long as they are recording your time effectively. Hypothetically there may be some legal issue regarding an employee using their own phone to call in, especially with cell phone minutes and the cost associated with them, but by itself this mechanism is not illegal.

It is perfectly legal, although it strikes me as a colossal waste of staff time, especially if there is a large number of employees. An employer must maintain a record of its employees' work times and can do so however it chooses. If the employees cannot reach the right person in the main office or reach voice mail and have to call back, etc., then the employees' work time must reflect a start time when the employee first tried to make the call, not when the employee finally succeeded. You can equate this to an employee having to wait in a long, long line to punch a time clock. The employer must not create a situation where employees are required to spend a lot of UNPAID time complying with a work rule. This is especially important if the employer docks employees' pay or disciplines employees for a short tardiness.