An employer may decrease hours work for business reasons. It may not do so in retaliation for any prohibited activity, conduct or behavior.
This is a difficult question to answer, as in each jurisdiction, the law is somewhat different. Generally, an employer may search a computer provided BY the employer even when the employee uses such computer for personal purpose. However, it is a gray area as to whether the employer can search a PERSONAL computer brought to the workplace. I would think that this should not be allowed
You may not be struck. Read the contract carefully. You may be able to terminate based on misrepresentation, fraud, etc., as to the ability of the manage to manage your career.
Sad, but unfortunately, an employer could terminate an employee even under a suspicion of wrongdoing. They do not have to prove you did something wrong. You are most likely an employee at will.
Don't sign anything you feel you should not sign. The employer must pay you for work done.