Qualified plans typically exclude employees under age 21 or employees who haven't yet been credited with 1000 hours of service. In addition, plans can specify specific classifications of employees who will be excluded. You have to read the documents to know your rights. The official plan is also available, although you might be charged a reasonable copying fee. Unless you are clearly not a participant, the plan must provide the documents.
Some plans do allow voluntary after-tax contributions.
Often, the plan might specify that no distributions are made until you reach a certain age, e.g., 65. That is permissible, as are provisions that delay distributions if you enter into competition with a competitor. Still, you need to examine the SPD to determine what your rights are. Employer contributions are taxable unless they are rolled over to either an IRA or another qualified plan within 60 days of receipt. This is determined by whether or not you and or your spouse are currently covered by a qualified plan and your income level.