@Anna Marcotte ,
I may not be the best person to answer your 3 questions as I'm also not the most experienced investor but I can give my take. I also went through a similar experience when I purchased my 1st property last summer.
I live in the seacoast NH where property values have skyrocketed. The town I live in any duplex is at least 750k or higher so that was unattainable for me. I decided to buy a place where I could house hack by renting out rooms instead (although you have to be willing to live with other people). I was able to take advantage of a first time homebuyer purchase and put less money into the house and was still be able to get it. I went this route to save more money for the rehab/maintenance needed as well as the fact I would need less money to save up for my 2nd property, because I don't necessarily need to invest in the town I want to live in (cheaper markets nearby).
A) The BRRRR method works the same either way if the numbers work. The only benefit of it being your primary house is you can put less money down to acquire the property. When you refinance out you can buy a new house to be your primary resident and take advantage of your second property being a residence as opposed to a investment property (turn your first house into a rental).
B) If your doing a primary residence you'll be better off using your own money as there is less people involved in the house you're currently living in. If you go the investment property route you have both options but it'll be tougher to find hard money lenders as a new investor and if you do they may be unfavorable terms.
C) I know people that have done LLC's on their primary residence but it was more as a liability shelter if they were renting out other units or rooms. I personally have it in my name and not an LLC. Although you should look into the terms of your mortgage because if you try to switch it later to an LLC some banks will have that set off the acceleration clause since it's technically changes names. I would be careful asking the bank about that as well because it will flag you as someone trying to be an investor but take advantage of a residential loan.
Like I said take my info with a grain of salt but that's what I've experienced / seen in my time in REI so far.