I have an old (1885) house with basement drains located in all of the 6 below ground rooms. The drain piping from the kitchen sink on main floor runs vertical and ends approximately 8" above a basement floor drain - resulting in an open air gap. This floor drain is a direct run approx 75' to the main city sewer line. Typically this floor drain keeps up with the sink drainage unless too much dish detergent is used and we sometimes get soap suds overflow onto basement floor. (The 22-yr old does the dishes) There is a second floor drain located approximately 10' away - piped directly to the other drain.
Q1: Any suggestions - cautions with extending the drain piping and eliminating the air gap?
Q2: Any suggestions or cautions with adding a dishwasher with discharge into the sink drain piping?
Q3: Could - should I measure the drainage rate and if so, what is an acceptable / minimum rate to add a dishwasher?

Do it the right way and eliminate the air gap.

That's what I'd like to do. But would there be a reason it wasn't installed that way? Are there any potential issues or concerns I should be aware of? Any issues with installing a dishwasher once I eliminate the air gap?

less than zero..

Absolutely tie it directly in. Badda boom badda bing. Done deal