K1- I would plan on it taking a total of 6-8 months until you have your visa in hand. Each case is different so theres no exact time length I can give you. I filed my K1 in september and had my initial approval 2 weeks ago (so about half way there now), however, there are people who filed in June, July and August who are still patiently waiting for their initial approval, therefore it will take longer than 6-8 months for these people.
Not too sure about the AOS thing as I wont be going through that until August.
You can visit during the K1 process, though you are right about the proof of ties to show intent to leave. Having a K1 in the system could mean that you will be extra scrutinised (this has happened to many, whilst many have breezed through without a problem). Proof of ties can be a letter from your employer stating when youre returning back to work, a letter from your landlord (tenancy agreement), mortgage papers, car loans, bank loans.
Hope that helps