The formal eviction process will hurt the tenant with being able to find future rental properties, so they should equally be motivated to avoid that.

The few times i've had something similar, I just used to the process to formally get them out and retake possession. 

or wait until the formal notice?

We (judiciously) extend credit and we pay for pest control and act quickly in the event of bedbugs.

If you prefer the formal process of going through a buyer's agent that early in the shopping process, let the agent know up front your timeline and your motivation or lack thereof for buying on that timeline. 

If there are multiple violations or they are egregious (unauthorized animal, trashing the property, etc.) then I will hit them with a formal, written warning.

I've worked in property management for 2 years now and it's really helped me gain a better understanding of the industry, and has been helpful studying for my broker's exam as I've encountered many examples of items I'm now formally learning about. 

Given that my ADU is attached to my primary residence, should I go through the formal process of filling out a rental application?

If you have concerns, I usually request a "City-Will-Serve" letter from the City in which they provide a formal letter stating that the City service has capacity to accept your added flows.

@Jacob Claxton formally your lenders would need to tell you if you qualify for such a thing but in theory, it is possible to do.