There were no "tickets" in the early days. The runner-up for President became Vice-President. This doesn't mean that the widespread charges of corruption were any truer then, nor caused by any higher motives than they are now. Various historians say that election was fraught with insane partisan bickering, and insane partisan journalism -- which became "history". Sound familiar?
Incorrect. Although the term was not used perhaps, the Democratic-Republican congressional caucus was very clear during the campaign that Jefferson was their candidate for President and Burr for Vice-President. That fits my definition of a ticket.
The runner-up for President became Vice-President.
Absolutely correct, in the original intent of the Founders. Which was a defect changed by the 12th Amendment, as it caused major problems in both the 1796 and 1800 elections. The original system was based on the assumption that "factions" would not develop in the country. With the rise of the party system, the original system was doomed.