Hey Clayton -
First off, congrats on using the 203k successfully on your first BRRRR/househack! That's awesome. I got started very similarly!
To answer your questions.
1. Your underwriter on your next loan is going to see what you're doing. Now, technically, you're moving out of your first, refinanced out, and no longer have the FHA. Lenders will definitely challenge you on this. This loan isn't meant to be a flipping loan you do over and over again, but if you have good reason and are playing by the rules, it can be done. Would it kill you to wait the year before your next one?
2. Again, you can use another 203k to go into another property. But, your underwriter will challenge you on this. Why are you moving so soon? Why would you go from one 3 unit, into another 3 unit for your own home you want to occupy? Does that follow conventional wisdom?
3. Yes, there are a few. The Fannie Mae HomeStyle, Freddie Mac Choice Renovation, and some other oddballs here and there. But those are the most popular products. Those loans generally are similar to the 203k. I'd say the main points that might concern you would be that the HomeStyle down payment requirements rise with the amount of units you want to buy. Unlike the 203k which is 3.5% flat across the units, HomeStyle ranges from 3% down for single family, up to 20-25% down required for 3-4 units.
I love your enthusiasm to roll right into the next one, but in my experience, just let time do it's thing. Many doors will continue to open for you if you make good deals.
You won't have to keep leveraging owner occupant loans to the point of flagging lenders from wanting to work with you.
You have some equity now. Look into combining that equity with some more conventional loan products, or hard money lenders, etc.
I absolutely love the 203k and the power it has, but ultimately, it is a logistically intensive process compared to other methods.
Best of luck either way, man. And congrats again!