When this happens, does the stove's breaker trip too?

Nope

Some higher power appliances (such as stoves) may have some functions on a separate 110 circuit. When the stove causes other circuits to trip, is everything on the stove still working?

Yes the stove still works, it just trips breakers in other areas of the house. Mainly just the living room. Usually happens when more than one burner is turned on.

Does the stove get its power from one 2pole breaker, or two separate breakers that are locked together, or, two separate breakers that are not connected to each other?

Range is on one two pole 40A breaker inside panel.

My range, which is a gas cooktop and an electric oven, has a 50A breaker for the oven alone, as well as the 20A 115 volt circuit for the controls.

it doesn’t trip the breaker for the range and there is only one 220 plug on my stove.

Did you notice if some lights flicker or dim while other lights get brighter while you are using the stove?

Is your living room breaker either an AFCI or GFI type?

Living room breaker is afci. No I don’t notice any lights dimming or changing when stove is on.

I will try that. Thank you