The Tenure of Office Act of 1867, not to be confused with the one of 1820, is what Andrew Johnson was accused of breaking. I think his removal of Sec. Stanton was a few months later, though the Senate was not in session until January of 1868 to deny confirmation. (Is it still called confirmation when approval to remove someone?) Pres. Johnson was acquitted by a single vote. You were right in saying the language is unclear. If my memory is correct, cabinet members got a one month protection from removal. The act was repealed and later declared unconstitutional by the Supreme Court in a different case.
On the election of Senators: It was the state legislators, not the House of Representatives. Nathan said it was the House, without specifying if it was a state or the federal one doing the appointing of Senators. David later implied that the appointment came from the states in his theory on state machinery, Corruption was a big reason for the 17th amendment. Another was lack of representation. Sometimes a state did not decide on who to send to the Senate and would lack one, and very rarely both, Senators. There were calls for direct election of Senators in the 19th century, including from Andrew Johnson (since I drove on his highway Tuesday, I might as well mention him in both paragraphs). William Jennings Bryan was the main proponent of the change in the 20th century. Fun fact, Maryland ratified the amendment earlier this year.