Before I answer this I want to state that most revenues for the Federal Government came from tariffs and excise taxes not from private donations, they still got private donations though. Tariffs were placed on imported goods and there were excise taxes on products such as alcohol or tobacco. As a result of this, these taxes burdened working class Americans (the majority) since they were forced to spend a larger portion of their income on goods compared to the rich that could comfortably afford to buy these things. By passing the 16th amendment, the burden of providing revenue to the Government switched from the poor to the rich. When your teacher says that private donations from corporations funded the Government he isn’t entirely correct. Yeah I’m sure there were a couple cases where they gave private donations but where the Government really made their money was when corporations imported goods for use in the US. The tariffs paid by corporations on imported goods was the main contributor.

The adoption of the 16th amendment was not there first time an income tax was implemented by the Federal Government. An income tax was first proposed in 1812, but saw its first implementation by Congress, under the Union, in 1861. This was to relieve the accrued debt of the Union caused by the Civil War. A progressive income tax was set in place. Those with an annual income between $600 and $10,000 were taxed at the rate of 3 percent while those earning over $10,000 paid 5 percent. The larger your income the more taxes you had to pay. This income tax was decreased following the end of the war and then abandoned in 1872. In 1894, under Democrat President Cleveland and a Democrat control of both houses, a 2% income tax was implemented on those earning more than $4,000 a year. Within a year of the income tax implementation, the SCOTUS deemed it unconstitutional in a 5-4 decision on the basis that income tax was forbidden under Article 1, Section 9 of the Constitution. Which is odd because the SCOTUS upheld the income tax during the Civil War.

At the turn of the century, progressive reform began to gain traction and support for the income tax came back. Since the SCOTUS had struck down the previous income tax, the only way to implement it would be to amend the Constitution. It was presented and approved by both houses of Congress in 1909 and was ratified by 42 states in 1913.

Most Americans thought the new tax was a great idea. One taxpayer wrote to the Bureau of Internal Revenue, "I have purposely left out some deductions I could claim, in order to have the privilege and the pleasure of paying at least a small income tax. . . .”

Today, private donations to the Federal Government (not including other departments and other functions such as the Smithsonian) are only about $3 million in total and they get thrown directly into the general fund, just like taxes. It has always been this way. Private donations have always been incredibly small compared to the money gained from tariffs and taxes.

Nearly all private donations were to campaigns and specific politicians.

As for you final question, the implementation of the income tax had no affect on corruption. Bribes were given directly to the politicians themselves and they used that money to fund their lifestyles or to do what they saw fit with it.