2 Answers

No, if you are the employee earning and receiving the wages, they should be reported under YOUR social security number.  
Your father may still claim you as a dependent; I suggest you talk to him about this arrangement.   

No. To report income under another person's SS# might be considered fraud. Your father is taking credit for your income toward his retirement. Your income could cause your father's income to incorrectly exceed the maximum taxable social security income. The IRS and Social Security Administration do not apply the No Harm No Foul principle. They see any false reporting as a crime. 

There are forms to file on a quarterly basis. If you do not file them with an estimated tax payment, you'll be fined at the end of the year and charged interest. The fine and the interest are not heavy, so it is much less an issue for not filing at all than to file falsely.

At the very least, put 20% of your income in the bank and save it for tax time. If you earn less than $3000 in the year (I'm guessing on the dependant number it is $5000 if your father is not counting you as a dependent), you won't need to file taxes. Don't forget to figure in tips. If you record your tip income daily, you'll be able to use that as a basis for taxes. If you do not, the IRS will estimate your tip income and hold you accountable to their estimate.